A111D2 M3bDtl

NAri INST OF STANDARDS & TECH R.I.C.

A1 11 02436061

Howvtt, Qdfald Leona/Llnear opponent>col OC100 U56 NO.85-3202 V1986 C.1 NBS-PUB- Linear Opponent-Colors Model

Optimized for Brightness

Prediction

Reference

NBS

PUBLICATIONS

Gerald L. Howett

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Gaithersburg, MD 20899

June 1985

Issued February 1986

~QC

100 sored by

.056 epartment of Energy

85-3202 ’

1986

MBS EESEARCH WFOSITATION CENTER G

NBSIR 85-3202

LINEAR OPPONENT-COLORS MODEL

OPTIMIZED FOR BRIGHTNESS

PREDICTION

Gerald L. Howett

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Gaithersburg, MD 20899

June 1985

Issued February 1986

Partially sponsored by

The U.S. Department of Energy

Washington, DC

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director
nikmu.

'". .'

: i<s7‘

& f -1^ 3a«9MWO:?' =»0:T»3MfjS>A«Ji3a5£tf;^

i$^f 9nwU

dau ^ bfoiitdar

i.r'V'i,

yt! t>»iu9rioqf y|f«rf^a«) _

Ve’»»^3 5P )ne»r»tiaqet? ,#,U 9t^T ?

^ " OiX .iwiittntrfgftW

^

TM3WTIf4^

.»a»«ir«t ,4UiMAeH«AY« >»C^'UAIlTl*i4l MW^ITAI^
MSTRACT

Formal multivariate optimization techniques were applied in an attempt to

determine how well a linear, opponent-colors model of color vision could account

for specific brightness-matching data. The data fitted were from a single

experiment by Sanders and Wyszecki that matched an adjustable white light in

brightness to each of a set of lights of 96 different colors and constant

luminance. A generalized, linear, opponent-colors model was formulated, which

Included the models of Guth (and coworkers), Ingllng (and coworkers), and

Thornton as special cases. The model contained 10 parameters, including nine

determining the spectral responses of the three opponent-level channels , and one determining the rule for combining the outputs of the three channels to

obtain an estimate of equivalent luminance (the luminance of an equally bright

white light). Despite difficulties with the optimization procedure, a model

was found that correlates better than 0.98 with the fitted data. The predictions of this model for various other color-vision functions were explored and

compared with corresponding predictions of the Guth and Lodge model and the

Thornton model. The new model's predictions of these functions are less than

perfect, but suprisingly good considering that the model was optimized entirely

on brightness data (the only restriction being that the luminance channel

should not have any negative values). The model was shown to predict the sort

of complex mixture of sub- and superadditivity that is present in actual data.

Some new algebraic results concerning the "B/Y” or ”B/L” (equivalent luminance

over luminance) ovals on the chromaticity diagram were derived.

Key words; additivity; brightness; color; equivalent luminance; Guth model;

luminance; model, color vision; opponent colors; optimization,

multivariate; primaries; white point.

ill
FOREWORD

This report is one of a series documenting National Bureau of Standards research

and analysis efforts in support of the Department of Energy/Lawrence Berkeley

Laboratory/National Bureau of Standards "Lighting Technology" Program, The

research described in this paper was supported partly by the U.S. Department

of Energy, through the lighting program of Lawrence Berkeley Laboratory, under

Interagency Agreement No, DE-AC03-76SF-00098 ; and also by the NBS-funded project

"Optimized Model for Brightness,"

The author acknowledges with thanks the stimulating critiques during the course

of the work provided by the Lawrence Berkeley Laboratory Technical Monitors,

Drs, Sam Berman and Robert Clear,

An oral report of portions of this study was presented to the Optical Society

of America at its fall 1985 annual meeting in Washington, D,C,

ACKNOWLEDGMENTS

The author is grateful to a number of members of the NBS Center for Applied

Mathematics for consultations regarding various mathematical aspects of the

work described in this report. These generous colleagues are in no way

responsible for any statements concerning mathematical matters, or any

equations, contained within the report.

In addition, particular thanks are due to Mrs, Donna Shoemaker, who managed

to do a difficult typing job very well, under difficult circumstances.

iv
TABLE OF CONTENTS

Page

ABSTRACT iii

FOREWORD iv

ACKNOWLEDGEMENTS iv

LIST OF TABLES viii

LIST OF FIGURES lx

A NOTE ON NOTATION; UNDERLINING xi

1. INTRODUCTION 1

1.1 HISTORICAL BACKGROUND 2

2. THE NUMERICAL MODELS 4

2.1 GUTH’S MODEL A 2.2 THORNTON’S MODEL 8

3. THE GENERALIZED LINEAR MODEL 8

4. OPTIMIZATION STRATEGY FOR THE GENERALIZED LINEAR MODEL 9

4.1 ERRORS IN THE PUBLISHED DATA 11

5. PROBLEMS WITH THE MINIMIZATION, AND THE BEST RESULTS OBTAINED ... 13

6. PREDICTIONS BY VARIOUS MODELS 18

6.1 PREDICTIONS OF THE SANDERS-WYSZECKI BRIGHTNESS DATA 19

6.1.1 Howett, Best-Fitting 19

6.1.2 Howett, Restricted 19

6.1.3 Guth and Lodge, Original 21

6.1.4 Guth and Lodge, Adjusted 24

6.1.5 Thornton, Original 26

6.1.6 Thornton, Adjusted 26

6.2 OPPONENT-LEVEL CHANNEL RESPONSES 29

6.2.1 Guth and Lodge 30

6.2.2 Thornton 30

6.2.3 Howett, Best-Fitting 33

6.2.4 Howett, Restricted 33

V
TABLE OF CONTENTS (Continued)

6.3 Page

SATURATION OF THE SPECTRUM 35

6.3.1 Guth and Lodge 37

6.3.2 Thornton 39

6.3.3 Howett, Best-Fitting 39

6.3.4 Howett, Restricted 39

6.4

EQUIVALENT LUMINANCE OF THE SPECTRUM 43

6.4.1 Guth and Lodge 44

6.4.2 Thornton 44

6.4.3 Howett, Best-Fitting 44

6.4.4 Howett, Restricted 44

6.5

WAVELENGTH DISCRIMINATION 49

6.6

NEUTRAL POINTS 54

6.6.1 Guth and Lodge 57

6.6.2 Thornton 59

6.6.3 Howett, Best-Fitting 61

6.6.4 Howett, Restricted 61

6.7

THE PRIMARIES 64

6.7.1 Guth and Lodge 65

6.7.2 Thornton 68

6.7.3 Howett, Best-Fitting 71

6.7.4 Howett, Restricted 75

6.8 CONTOURS OF CONSTANT B/A 75

6.8.1 The Special Case p=2 (Including the Guth Model) 78

6.8.2 The Special Case p=l (Including the Thornton Model) . 79

6.8.3 The General Case (Including the Howett Model) 80

6.8.4 Guth and Lodge 83

6.8.5 Thornton 83

6.8.6 Howett, Best-Fitting 86

6.8.7 Howett, Restricted 89

6.9 ADDITIVITY 89

6.9.1 The Significance of Nonadditivity 89

6.9.2 Vectorial Considerations 92

6.9.3 Definition of a Generalized Nonadditivity Index 92

6.9.4 The Relationship of the Power, to Nonadditivity .. 94

6.9.5 Actual Predictions: Pairs of Monochromatic Lights .. 96

6.9.6 Actual Predictions: Lines of Constant Dominant

Wavelength 96

Vi
TABLE OF CONTENTS (Continued)

Page

7. SUMMARY AND CONCLUSIONS 102

7.1

SUGGESTIONS FOR FURTHER WORK 105

7.1.1 Some Proposed Experimental Studies 105

7. 1.1.1 Is Chromatic Brightness Visually Effective? . 105

7. 1.1.2 How Much Coloration in a Light Source Is

Tolerable? 106

7. 1.1.3 The Effect of Viewing Conditions on Equivalent Luminance 106

7.1.2 Some Proposed Computational Studies 107

7. 1.2.1 Individual Observer Variation 107

7. 1.2.2 Studies of Nonlinear Models 107

7. 1.2. 3 Further Predictions of Color-Vision

Functions by The Optimized Linear Model

of This Report 107

8. REFERENCES 108

APPENDIX: BASIC COLORIMETRIC QUANTITIES AND NOTATION 109

CAPITAL LETTERS 109

LOWER-CASE LETTERS 110

LOWER-CASE LETTERS WITH OVERBARS Ill

vii
LIST OF TABLES

Page

Table 1. Null Lines and Primaries of Opponent-Level Channels; Guth

and Lodge Model 69

Table 2. Null Lines and Primaries of Opponent-Level Channels:

Thornton Model 72

Table 3. Null Lines and Primaries of Opponent-Level Channels:

Best-Fitting Howett Model 74

Table 4. Null Lines and Primaries of Opponent-Level Channels:

Restricted Howett Model 77

viii
LIST OF FIGURES

Page

Figure 1. Correlation plot for best-fitting Howett model 20

Figure 2. Correlation plot for restricted Howett model 22

Figure 3. Correlation plot for Guth and Lodge model 23

Figure 4. Correlation plot for adjusted Guth and Lodge model 25

Figure 5. Correlation plot for Thornton model 27

Figure 6. Correlation plot for adjusted Thornton model 28

Figure 7. Opponent-level channel responses for Guth and Lodge model .. 31

Figure 8. Opponent-level channel responses for Thornton model 32

Figure 9. Opponent-level channel responses for best-fitting Howett

model 34

Figure 10. Opponent-level channel responses for restricted Howett

model 36

Figure 11. Saturation of the spectrum, predicted by Guth and Lodge

model 38

Figure 12. Saturation of the spectrum, predicted by Thornton model .... 40

Figure 13. Saturation of the spectrum, predicted by best-fitting

Howett model 41

Figure 14. Saturation of the spectrum, predicted by restricted

Howett model 42

Figure 15. Equivalent luminous efficiency function, predicted by Guth

and Lodge model 45

Figure 16. Equivalent luminous efficiency function, predicted by

Thornton model 46

Figure 17. Equivalent luminous efficiency function, predicted by

best-fitting Howett model 47

Figure 18. Equivalent luminous efficiency function, predicted by

restricted Howett model 48

Figure 19. Wavelength discrimination function, predicted by Guth and

Lodge model 50

ix
LIST OF FIGURES

Page

Figure 20. Wavelength discrimination function, predicted by Thornton

model 51

Figure 21. Wavelength discrimination function, predicted by bestfitting Howett model 52

Figure 22. Wavelength discrimination function, predicted by restricted

Howett model 53

Figure 23. Opponent-channel null lines and white point, implied by

Guth and Lodge model 58

Figure 24. Opponent-channel null lines and white point, implied by

Thornton model 60

Figure 25. Opponent-channel null lines and white point, implied by

best-fitting Howett model 62

Figure 26. Opponent-channel null lines and white point, implied by

restricted Howett model 63

Figure 27. Opponent-level primaries of Guth and Lodge model 66

Figure 28. Opponent-level primaries of Thornton model 70

Figure 29, Opponent-level primaries of best-fitting Howett model 73

Figure 30, Opponent-level primaries of restricted Howett model 76

Figure 31, Constant-B/A loci, predicted by Guth and Lodge model 84

Figure 32, Constant-B/A loci, predicted by Thornton model 85

Figure 33. Constant-B/A loci, predicted by best-fitting Howett model .. 87

Figure 34. Constant-B/A loci, predicted by restricted Howett model .... 90

Figure 35. The principle of vector addition 93

Figure 36. Additivity of mixtures of two monochromatic lights,

predicted by restricted Howett model 97

Figure 37. Additivity along lines of constant dominant wavelength,

predicted by restricted Howett model 100

X
A NOTE ON NOTATION: UNDERLINING

In standard printing practice, letters representing mathematical variables are

printed in italics to distinguish them from ordinary text. Since this report

is photoreproduced from a typed manuscript, the customary practice of underlining

to represent italics has been followed. To avoid the unnecessary visual clutter

of too much underlining, variables that have other letters or symbols attached

to them, either vertically or laterally, have, in almost all cases, not been

underlined. (This includes variables having subscripts, superscripts, or

overbars; or which appear as parts of equations or algebraic expressions.) In

short, the intention was to underline only single, unadorned, isolated letters.

Underlining in this report never has any significance other than simply indicating

italics. In particular, vectors are represented by capital letters with

overbars, and no matrices or tensors are used at all. In no case does an

underlined letter differ in meaning from the same letter appearing without the

underline, even where the above rules appear to have been violated. (There are

almost surely examples in the report of failure to adhere rigidly to these rules.)

All diacritical marks other than underlines are used to distinguish different

quantities. Likewise, capital letters always represent variables distinct from

those indicated by the corresponding lower case letters. To Interpret these

distinctions, see the Appendix, which summarizes standard colorimetric notation.

xi
r^v, m m*i a

***“

'- = - - - * a

‘S-,- ;*’*^**",'. , ff-..

''-I-, /*"• ^^ »(.:> i.fc.JS'

‘ ‘'^

ba?«;mi,.

; W(; a -P’-y

«J toj

^m^3^Q3

mit- ' ^!l eT^M^isrro;

%^mm^-i’’:ity.xi \l(io dntlT*i»«» cw“%iw n&i3n»:^i'‘.4^

•*" ^fjt' .. ••<•>•»« #4il%'Alt :!% j»,A'«=il^» s •_ Wt

,

anijegfeji f^^^..m^»lll^frn^u';m»•'fi^ H

*»tiju-T.fc»o!

ft =..;_ '

rfiiw ' *'xi ' 84did4y ‘

, If ;ei»tilfjf

saki-*' • ;«» a

^

Q.

l:;V'''''-^ srffb tofts nrr^-^'lllfSmfl^iti t»1[ttfc*1®3j&l*64ai4l£HOT'll

yyj,-

:i6 »iA im^tri »ydlr* »il^ »rta^*niw4^iijRll7iLa4i

( w sMib^ 'fftoiB* Hi i^’*' *K5^ tt ' ^Jii o» ''5

>/yo^«fff|t> ^«r]0S^i7Affc*ftl^«»Aci »ij|" •iBj»iil’#l4rn( TtS*fe\:!^tik* iBsill'^aAll^ ftk

i ' iftTa iil ^'«v®2^'*iSjiir ^filtUo^Si??oo®!5J saoSs

.jfi #^i%iL*R>»

*^

htv ,Hil(a9q4^ 'sM^^enoiiooJrsItlb

Unhung ,

» .» 4%.fH0-* , r f . . . « *^ » » f.r « . . i .

.

'•^ • i > •*«P 4 |>. I »*«<•' . .. *• *i)tr

tv’-

^A.-1'J
1 . INTRODUCTION

It has been known for a long time that if a strongly colored (saturated*)

light, such as a red, for example, is examined side by side with a white light

that measures the same luminance as the red on a light meter, most people will

say that the red appears to them to be considerably brighter. At one time, it

was said that the cause of this phenomenon (the Helmholtz-Kohlrausch effect)

was that people tended to confuse saturation with brightness.

The predominant current view is that the red looks brighter than the white of

equal measured luminance not because of any confusion in people's minds, but

because the visual system is organized in such a way that the red really does

stimulate the brightness-perceiving center of the brain more than the white

does. The class of theory that explains this effect, at least in a qualitative

way, is a version of opponent-colors theory. [The essential details of this

class of theory are listed in section 1.1. Also, see Howett (in press) for an

extended explication of the problem.]

The simplest version of the class of opponent-color theories that explain the

Helmholtz-Kohlrausch effect are the linear models. Until now, these linear

opponent-color theories have been somewhat successful at explaining additivity

problems related to the Helmholtz-Kohlrausch effect, at near-threshold luminance

levels, but less successful at modeling these effects at suprathreshold levels.

Since most vision in everyday life takes place at luminances well above threshold,

* Saturation is a perceptual variable, indicating, for lights, the perceived

degree of difference of the color of the light from an equally bright white

light. It can be identified with "strength” or "vividness" of the color.

There is no easily specifiable physical correlate of saturation.

1
photometrists are actively seeking a formula for predicting, with some accuracy,

the brightness relations among suprathreshold lights (CIE, 1978)*

The present Investigation was directed toward deriving such a formula and

seeing how well It could predict at least one particular set of experimental

results. The basic approach was to write a formula for a generalized version

of the linear opponent-colors model, and then to optimize the constants In the

formula to obtain the best possible agreement with the behavioral data.

1.1 HISTORICAL BACKGROUND

S.L. Guth has been using a color-vision model of the opponent-colors type to

explain why luminance (measured) and brightness (perceived) can disagree strongly

when the lights being compared have different chromatlcltles. The essential

features of Guth's model, which he has elaborated In a series of publications

for about the last 20 years, are as follows:

(a) The spectral absorptance curves of the three cone pigments can be

expressed as linear combinations of the CIE 1931 color-matching

functions, x(X), y(X), and T(X)*. [Most vision researchers accept

Judd's (1951) short-wave revision of these functions as closer to

correct, but Vos (1978) has shown that Judd's curves can themselves be

well approximated by a linear combination of the CIE functions.]

(b) The spectral response functions of the three opponent-color processes

(of which two are opponent and chromatic, and the third is the

* A brief description of the basic quantities of colorimetry and the standard

notation Is given In the Appendix of this report.

2
non-opponent luminance channel) arise from reassortment of the outputs

of the three cone types in three different combinations. The contribution of each cone output to each of the opponent-colors channels

may be either excitatory (positive) or inhibitory (negative).

(c) It is assumed that no nonlinear processes intervene between the

absorption of light within the cone pigments and the generation of

output within the opponent-colors channels. It is in this sense that

the model is "linear". As a result, the outputs of the opponentcolors channels are themselves linear combinations of the CIE 1931

color-matching functions.

(d) Guth then assumes that the outputs of the opponent-colors channels

are further processed as if they were independent dimensions of a

three-dimensional euclidean vector space. In Guth’s own terms, his

model is a vector model. Any color can then be thought of as a vector

from the origin, with its components equal to the outputs of the

three opponent-colors channels.

(e) The length of this color vector, calculated by the usual euclidean

rule —square root of the sum of the squares of the components —is

called "vector luminance" by Guth, and is identified by him as a

measure that does agree with brightness judgments, regardless of

chromaticity , at least under conditions at or near threshold

detection.

A model very similar in structure to Guth’s, differing only in its quantitative

details, has been under development by Ingling and his coworkers (Ingling and

Tsou, 1977; Ingling, 1977) for a comparable period of time. Since the Ingling

3
model introduces no new basic features for purposes of the present investigation, only minimal further reference will be made to it. This omission

should not be construed as reflecting at all on the relative merits of the

Guth and Ingling models.

Later, Thornton (1973) accepted the basic concepts of Guth’s model, but changed

the opponent channel responses, as well as the final formula for combining the

components of the color vector. Thornton reasoned that nerve cells tend to

combine their outputs in a direct fashion, in which the total stimulation of a

cell is simply the sum of the stimulations received from the input channels.

He therefore took the metric of his vector space to be simply the sum of the

absolute values of the components (the opponent-colors channel outputs), in

contrast to the square root of the sum of squares used by Guth and by Ingling.

2. THE NUMERICAL MODELS

2.1 GUTH'S MODEL

Various versions of the Guth model have been published by Guth and an assortment

of coworkers over the years. In a recent version (Guth, Massof, and Benzschawel,

1980), a great deal of flexibility in the form of six adjustable constants was

introduced. These constants represent allowances for the process of chromatic

adaptation as applied separately to the cone responses and to the opponentlevel channel responses. Although adjustments for adaptation must undoubtedly

be part of a complete color-vision model, the Guth-Massof-Benzschawel treatment

does not explicitly express these parameters as functions of luminance,

chromaticlty , or other measurable variables; they are in practice arbitrary

constants to be fitted to data by trial and error. Since there was no obvious

basis for selecting any particular values for the adjustable constants, the

4
version that follows was chosen from an earlier publication (Guth and Lodge,

1973), in which the model still had a fixed format.

For purposes of this analysis, there was no need to separate steps (a), (b),

and (c) of section 1. Mathematically, applying a linear transformation to

another linear transformation yields still another linear transformation.

Therefore, the step referring to the cone responses is omitted, and the

formulas for the opponent-colors channels are the starting point of the treatment. These formulas, for the Guth and Lodge (1973) model, are:

a = 0.954y+0.010z

t =

d = 0.799X-0. 6462-0. 167^-0.058y+0.030z.

(la)

In Eqs. (la), x, y, and z are the CIE 1931 color-matching functions, and are

functions of wavelength, as are the opponent-colors responses a, t, and d

(Guth's notation). Thus, x, for example, would be more explicitly written as

x(X), but the conventional use of overbars to denote color-matching functions

(formerly also called spectral distribution coefficients) makes it unambiguous

to omit the parenthetical ^ in most contexts.

In applying the model to specific colors, x, y, and z are replaced by the

general tristimulus values X» Y, and on the right sides of Eqs. (la), and

a, t, and d are replaced by tristimulus values A, T^, and ^ on the left sides.

The resulting equations are

A = 0.954Y+0.010Z

T = 0.799X-0.646Y-0.167Z (1)

D = -0.058Y+0.030Z

B = (a2+t2+d2)1/2.

5

( 2 )
In Eqs. (1), kf T^, and D are no longer functions of wavelength but apply to the

single color of which _X, Y, and ^ are the tristimulus values. It is not a

coincidence that the coefficients in Eqs. (1) and (la) are identical. The

quantities x, y, and z of Eqs. (la) are themselves tristimulus values, of the

monochromatic lights comprising the visible spectrum (with equal energy at

all wavelengths). In fact, the currently approved name (CIE, 1970) for these

quantities is "tristimulus values of the spectrum." The same quantities are

also properly referred to as "color-matching functions" because [see the

Appendix], in any colorimetric system, the tristimulus values of the spectrum

determine what light spectra match each other visually.

Since Eqs. (1) are fully general, they apply to any color; in particular, they

apply to monochromatic lights. Thus, Eqs. (la) apply to a subset of the stimuli

to which Eqs. (1) apply, and hence the two sets of equations must have the same

coefficients. This is a general rule, applying to transformations between any

two colorimetric systems that have a linear relationship to each other. As a

result of the validity of this rule, the two sets of equations, (1) and (la),

are thought of as equivalent and are often used interchangeably in publications.

It is then left to the context to communicate to the reader whether equations

having the form of (1) are intended to refer to spectral functions or the

tristimulus values of individual colors.

In Eqs. (la), a, t, and d represent the spectral outputs of the opponent-colors

channels. The ^ channel [the more common notation based on Eqs. (1)] is nonopponent and is the luminance, whiteness, or achromatic channel. The T^ channel

is usually described as the red-green opponent channel, in which red colors

(high JC) yield positive stimulation (T>0), and green colors (high jf) yield

6
negative stimulation (T<0). Similarly, the ^ channel is usually described as

the blue-yellow channel, in which blue colors (high yield (in Guth*s treatment)

positive stimulation (D>0), and yellow colors (high _X and _Y) yield negative

stimulation (D<0). The choice of sign in constructing these channels is arbitrary,

and other authors prefer to make yellow the positive phase of the ^ channel.

The quantity _B in Eq. (2) is, in Guth’s terms, the vector luminance because it

is derived from his vector model; ^ is itself a scalar quantity. ^ is not

Guth's notation, but that letter is commonly used in other publications on this

subject. It is often referred to as "brightness,” but is more accurately

described as a measure that correlates well with brightness. Brightness is

itself a subjective sensation level (CIE, 1970). Operationally, the _B value

of a light can be identified as the luminance of a reference white light that

matches the light in question in brightness. The name used for ^ in this report

is "equivalent luminance" (CIE, 1970), following the convention introduced by

the author in an earlier publication (Howett, in press).

In the current system of photometry, luminance is determined entirely by the

value of _Y, or, in the Guth model, the near-equivalent A. The essence of the

explanatory power of Guth*s model is that the quantity ^ includes not only the

luminance A, but also contributions from the chromatic (and opponent) channels ^ and JD. The result is that strong (saturated, vivid) colors which have

significant values of TT and/or have their equivalent luminances (El^) enhanced

by these contributions from the chromatic channels. This extra contribution

is often referred to as "chromatic brightness," and is a feature of virtually

all contemporary opponent-colors models, linear or nonlinear.

7
2.2 THORNTON'S MODEL

The following model is due to Thornton (1973). It is presented using notation

consistent with Eqs. (1) and (2), rather than in Thornton's notation. The

tristimulus-value version is given.

A = Y

T = 0.4X-0.4329Y+0.2073Z (3)

D = 0.4X-0.4322Y-0.2229Z

B = |A|+|T|+|D|. (4)

The nonopponent A channel is always positive, so that the absolute value sign

around A in Eq. (4) is not really necessary.

Aside from the difference in the rule for computing Thornton has definitions

of his opponent channels (T^ and quite different from those used by Guth.

The derivation of these definitions arose from some different considerations

in Thornton's treatment than in Guth's analysis, but here we are concerned only

with the mathematical form, and not, at this stage, with the theoretical interpretation.

3. THE GENERALIZED LINEAR MODEL

There are obvious similarities between the structures of the models embodied in

Eqs. (l)-(2) and Eqs. (3)-(4). First, Eqs. (1) and (3) both have the form

A = ai 1 13^

T “ a2i Xfa22Y+a23Z (5)

D »= a3i»-a32Y+a33Z,

where the aj[j (i,j = l,2,3) represent nine general constants.

8
Slightly less obvious is that the composition rules for ^ given by Eqs. (2) and

(4) are also of the same basic form. Both are special cases of the equation

B - (|a|p+|t1p+|d|p)1/p, (6)

where £ is a general constant. The metric represented by Eq. (6) is well known

in mathematics, and is often referred to as the Minkowski Lp metric. For Guth

[Eq. (2)1, p “ 2; and for Thornton [Eq. (4)], p

« 1.

It is necessary to take absolute values of the channel outputs in Eq. (6)

because fractional values of £ are possible, and these cannot in general be

applied as exponents to negative numbers. There was no need for the absolute

value signs in Eq. (2), since squaring a number performs the equivalent function

of making the sign of the number irrelevant.

4. OPTIMIZATION STRATEGY FOR THE GENERALIZED LINEAR MODEL

The model represented by Eqs. (5) and (6) contains 10 arbitrary constants

(parameters): the nine a^j values of Eq. (5), and £ of Eq. (6). The goal of

this study was to select a body of heterochromatic brightness'matching data and

optimize the 10 constants of the model, in the sense of making the predictions

of the model concerning equivalent luminances agree best with the data.

Two strategies were available in selecting a data set to work with. One was to

average data from several different studies; the other, to choose one extensive,

trustworthy, internally consistent body of data. The trouble with the various

significant heterochromatic brightness-matching studies in the literature —perhaps 8 to 10 in number— is that they were all carried out with different

viewing conditions (e.g., area, luminance, separation, presentation time). As

a result, although the general pattern of the results is similar in all the

9
reliable studies, the sizes of the numbers are very different. The author’s

opinion is that averaging such disparate data tends to obscure rather than

reveal the underlying structure of the situation. Therefore, the option chosen

was the single-experiment strategy, with the hope of successfully predicting

the data for one particular set of viewing conditions. An experimental study

that it seems important to carry out in the future is the systematic exploration

of the effects of varying these viewing parameters separately and in combination.

The experiment chosen was one carried out at the National Research Council of

Canada (Sanders and Wyszecki, 196A). Ninety-six colors scattered over the

chromaticity diagram, all having a fixed luminance of 20 cd/m^, were matched

in brightness by adjustment of a reference white light, by 20 observers. The

luminance of the matching white light —i.e., the equivalent luminance of the

colored light — was the dependent variable of the experiment. For each of the

colored lights , the experimental data can be embodied in three numbers : the

physically measured _X and 7^ tristimulus values (jf being constant at 20), and

the psychophysically (experimentally) determined equivalent luminance, B*.

What was optimized, in deriving the best-fitting model, was the agreement

between the empirically determined values, B*, and the model predictions,

of Eq. (6). Specifically, the goal was to minimize the sum of the squared

prediction errors.

96

S = 1 (B.-B*)2. (7)

i=l

The plan was to find those values of the aj[j in Eq. (5) and the value of in

Eq. (6) that minimized the objective function, of Eq. (7).

10
4.1 ERRORS IN THE PUBLISHED DATA

Several errors in the published tables of the Sanders and Wyszecki (1964) paper

have been uncovered. Three of these are typographical errors in the chromaticity coordinates of the samples. Since the tables contain CIE 1960 vi,jv

as well as x»2. values, there was no difficulty in deriving the correct 2. values,

on the assumption that there were not also errors in ^ or v. Because all the

colors used had chromaticitles lying along a series of straight, radial lines

in the chromaticity diagram, it was also possible to closely estimate the

value of ^ by linear interpolation between the coordinate of the two points on

either side on the same line, using the 2L values to determine the proportion.

The values of y^ derived from v and from linear interpolation agreed well

in all three cases.

The following table shows the correct and incorrect values;

Dominant Wavelength Published Correct

/Purity Level y Value y Value

It can be seen that in all three cases, the nature of the error is the transposition of digits, or the repetition of the wrong digit. It is easy to

visualize how such errors can be made in the process of setting type from the

original material supplied by the authors* The other group of errors (found by

Robert Clear, Lawrence Berkeley Laboratory), is more difficult to explain. The

published mean logarithm of what Sanders and Wyszecki refer to as B/Y values

(equivalent luminance divided by luminance) is in two cases not actually equal

to the mean of the 20 logarithms for the individual subjects. The following

table shows these discrepancies.

583/4

574/3

450/4

0.134

0.422

0.277

0.413

0.442

0.227

11
Dominant Wavelength

/Purity Level

Published Mean Correct Mean

Log B/Y Value Log B/Y Value

583/5

487/1

0.058

0.018

0.068

0.012

In both of these cases , the error Is In a single digit and could also be the

result of careless typesetting, but the source of these errors is somewhat

uncertain. One defensible reaction to this problem would be simply to omit

these two colors from the analysis. This Is the conservative strategy that is

above reproach. On the other hand, these dif f icult-to-obtaln data should not

be discarded lightly. In order for the mean logs to be off by as much as they

are, it is probable that errors in more than one of the individual values would

have had to be made, since no one individual value Is obviously grossly deviant.

Moreover, in contrast with the printed table of mean values, the published

table of the Individual log values Is clearly a photocopy of a typed table that

presumably was calculated and checked by the authors of that paper before It

was submitted. Consequently, the chances are good that the error was made In

the typesetting of the mean values, and not In the specification of the Individual values , so that the choice of adopting the actual means of the Individual

values as correct Is also defensible, from a probabilistic standpoint.

Actually, It most likely makes little difference whether a few points are omitted

from the analysis, or even whether a few slightly (but not grossly) erroneous

points are Included. This Is because the spread (variance) of the data Is

considerable, and because—as will be described later—the optimization of the

fitted model Is so difficult.

12
5. PROBLEMS WITH THE MINIMIZATION, AND THE BEST RESULTS OBTAINED

Although the optimization problem posed by the set of equations (5)-(7) appears

deceptively simple in form, it has proved to be extraordinarily difficult in

practice. The first attempt was made using an optimization routine included in

a commercial package of computer subroutines, usually regarded as of high

overall quality. Ideally, in applying an optimization procedure, one hopes

that the same optimum point will always emerge, regardless of the starting

point, the initial step size, and the other parameter values that can be varied.

Such behavior instills confidence that the true global minimum has been found.

It quickly became apparent that the situation with this minimization problem

was very far from this ideal. In fact, a new alleged minimum point (in

10-dimensional space) emerged with virtually every different choice of starting

parameters. It appears that the function ^ of Eq. (7) has a tremendous number

of local minima. As the starting point and initial step size were varied,

values of ^ lower than any previously discovered would turn up occasionally.

It became clear that it would be difficult to find the overall, global minimum,

and that even if , by chance , the global minimum were located , it would be

difficult to conclude that it was in fact the true global minimum.

Since the package routine was not succeeding, the present author decided to

program an optimization algorithm of his own that was of a conceptually simple,

brute-force type. It seemed, in advance, that the algorithm might require many

iterations, but that it might succeed in getting to the global minimum where

the more sophisticated package routine had failed. Unfortunately, the result of

applying the new program was very similar to the outcome of the attempt using

13
the package routine: a sequence of many different minima, with a new, lower

minimum emerging from time to time through trial and error.

A team of two mathematicians with special skills in optimization theory was

consulted regarding the difficult behavior of this particular function. They

were unable to find a global minimum even with the most advanced algorithms

available to them, and Indeed did not locate a local minimum as good as the

best one the author had found in his repeated attempts. Apparently, this

particular problem presents some sort of exceptional and unexpected challenge

to existing optimization theory and techniques. One factor that complicates

matters is the use of absolute values, which leads to sharp "points" or "spikes,"

rather than the more tractable smooth "domes" (in 10 dimensions). Another

possible contributing factor is that there may be interdependencies among the 10

variables, so that the number of independent parameters in the true underlying

problem may be less than 10; l.e., the present formulation of the problem may

be overdetermined.

Although for the above reasons a definitively best set of values cannot be

presented here, the best local optimum that the author has found predicts the

brightness data very well. It might really be the global optimum, or possibly

a slightly lower minimum will some day be found. In any event, the formula

presented below does establish two key points, which may be regarded as the

principal contributions of the present study; (a) it is possible to predict a

single body of heterochromatic brightness-matching data quite well with a

"linear" opponent-colors model; and (b) the use of formal, multivariate

optimization procedures can —in combination with some trial and error, as

14
required— produce useful and even surprising results, as opposed to the intuitive, pure trial-and-error procedures that have often been used in the past.

The best-fitting prediction of the Sanders-Wyszecki (1964) data that was found

is given by the following formula:

A = -0.0647X+1.0583Y-0.1294Z

T = 0.2914X-0.2341Y-0.0312Z (8)

D = -0. 1028X+0. 1572Y-0. 1639Z

P = 0.8184. (8a)

The quantity £ of Eq. (8a) is used to predict equivalent luminance, in

accordance with Eq. (6). The sum of the squared errors of prediction,

from Eq. (7), is only 36.047 for the 96 colors. The mean absolute prediction

error for a single color is 0.480, or 1.938 percent of the overall mean

value. The product-moment correlation, is 0.986.

It is important to note that the formula embodied in Eqs. (8) and (8a) was

obtained without the imposition of any restrictions whatever. The sole criterion

was minimizing the sum of the squared errors of prediction of the experimentally

determined equivalent luminances. However, despite the determination of the

formula exclusively by brightness data, Eqs. (8) represent a set of opponentcolor functions that resemble considerably the functions presented by other

authors. This promising finding encouraged the hope that a slight modification

of Eqs. (8)-(8a) might permit reasonable predictions of some of the classical

color-vision functions, such as the predictions exhibited by Guth and co-workers.

The most obvious defect of Eqs. (8) —as will be shown in a later section— is

that the luminance function A is negative in the shortwave (blue) extreme of

the spectrum. A valid luminance function cannot be negative anywhere, since

pure light of any wavelength (or mixture of wavelengths) at which the function

15
is negative would have to have negative luminosity, a possibility that experience

establishes as unrealistic. A second optimization was therefore carried out

subject to the single restriction that A be nowhere negative. In practice,

this restriction was simplified computationally to the triple limitation that

all three of the coefficients in the linear expression for a as a function of

X, y, and z must be non-negative. This guarantees an all-positive a function,

but the condition is excessively restrictive; i.e., all-positive a functions

can exist in principle even when the coefficient of x or z is slightly negative.

The restriction to a nowhere-negative luminance function is, at least on the

surface, still confined to brightness-related considerations. Nevertheless, as

will be seen later, the model that emerges is not too bad at predicting colorrelated functions.

The best model found under the restriction described above was:

A = 0.0015X+0.9865Y

T = 0.2691X-0.2056Y-0.0431Z (9)

D = -0. 0606X40. 0858Y-0.0582Z

p

= 0.8174. (9a)

The model embodied in Eqs. (9) and (9a) is not extremely different from the

totally unrestricted model incorporated in Eqs. (8) and (8a). Despite the

restriction imposed on a, this model predicts the brightness-matching data

nearly as well as the unrestricted model does. The sum of the squared prediction errors is a little higher, at 43.635; but the mean absolute error is

actually less, at 0.470 (1.896 percent). The correlation is 0.983, negligibly

lower

.

It should be pointed out that the coefficient of _X in the equation for A given

in Eqs. (9) is quite small. During the computer run that led to the particular

16
set of (at least locally) optimum values shown in Eqs. (9) and (9a), the

coefficient of ^ in the expression for A stayed equal to zero (meaning that it

was trying to go negative), but the coefficient of X kept a slightly positive

value. Nevertheless, there would probably be little consequence numerically

if the X term were simply dropped and A set equal to Y. A desirable step,

which has not been carried out, would be to make A proportional only to Y^,

(l.e., to choose an = aj 3 = 0 in advance), and then re-optlmlze the resulting

8-variable function.

The calculations of the predictions of this model, to be described later, were

based on the values as given in Eqs. (9). In a few cases, the predictions were

recomputed with the X term of the equation for A (i.e., aj]^) set equal to zero,

and the differences in the predicted curves were not noticeable, but not all of

the predicted functions were checked in this way. There is a statistical test

which can be applied to determine whether the predictions of equivalent luminance

made with Eqs. (9) are or are not significantly better than the predictions made

by setting an =0. Such tests were not carried out, however, because:

(a) the statistics are based on theoretical considerations properly applicable

to linear least-squares problems, and the degree of validity of these tests as

applied to nonlinear least-squares (as this problem would be classified, even

though the model is labeled "linear”) is not well understood; and (b) there is

so much "fuzziness" in the data and in the optimization process that expending

computational effort on statistical tests did not seem justified, in comparison

with using the time to explore another predicted color function. In the future,

it would be of interest to explore the use of statistical tests for deciding

between models that fit data almost equally well.

17
The most important result of the two optimizations [Eqs. (8)~(8a) and (9)-(9a)]

is that accurately predicting brightness data requires a power, considerably

less than unity; in both cases, 2, about 0.82. Thus, with respect to the

power in the combination rule, Thornton (p=l) was closer to the truth than

Guth (p=2), but even Thornton did not go far enough down to obtain a good fit.

6. PREDICTIONS BY VARIOUS MODELS

This section is devoted to showing, through a series of graphs, the predictions

by a group of models with respect both to brightness and to color-perception

functions. The prediction of color functions involves additional assumptions

beyond the specification of the opponent-level channel responses. It is

necessary to postulate, for each perceptual function to be predicted, a specific

mathematical rule for combining the opponent-level responses so as to match the

desired response curve reasonably well. Rather than Introducing new combination

rules tailored to make the predictions of the author's restricted model [Eqs.

(9)-(9a)] look as good as possible, the decision was made in advance to accept

without modification Guth's formulations of how to predict the various functions.

This procedure gave the advantage to the Guth model in the competition among

the models compared, with respect to the prediction of color functions. It was

felt that if the author's model could make a creditable showing under this sort

of restriction, there would be hope for its potential for providing a general

color-vision model, with possible further adjustment. In the present paper, no

modifications of the model are explored, but some comments are made regarding

new channel-combination formulas that can be examined in the future for possibly

improved predictions of some of the color functions.

18
6.1 PREDICTION OF THE SANDERS-WYSZECKI BRIGHTNESS DATA

This section presents correlation (scatter) diagrams showing predictions by

various mathematical models (ordinate) against the experimentally obtained

values for the Sanders-Wyszecki (1964) study (abscissa). It should be recalled

that the experimental values ["equivalent luminances," in the author's terminology

(Howett, in press)] are luminances of a white light seen as having a

brightness equal to that of each of the 96 colors tested, all of the latter

having been presented at a fixed luminance of 20 cd/m^ (=5.8 fL).

Thus, the vertical placements of the points in the scatter plots are a function

of the particular model, whereas the horizontal placements are fixed, in

accordance with the experimental results. If current photometry were correct —i.e., if luminance were perfectly correlated with brightness— then all the

points in each diagram would lie along a vertical line at an abscissa value of

20. Instead, in a now well-established pattern, 20 is the lower bound of the

equivalent luminances, and, for this particular experiment, the values range up

to approximately 37.

6.1.1. Howett, Best-Fitting

Figure 1 shows the author's best-fitting model, as embodied in Eqs. (8)-(8a).

As previously mentioned, the model predicts brightness very well (product-moment

correlation r=0.986), but involves an unrealistic feature: a luminance function

that contains negative values for a range of short wavelengths. The latter

feature will be exhibited later, in figure 9.

6.1.2 Howett, Restricted

The re-optimization [see Eqs. (9)-(9a)j was restricted to having an all-positive

luminance function. The prediction of brightness in the Sanders-Wyszecki

19
PREDICTED

VALUES

(cd/aXX2)

EXPERIMENTAL VALUES (cd/axx2)

Figure 1. Correlation plot (scatter diagram) for best-fitting Howett model.

Predicted equivalent luminance is plotted against the equivalent

luminance measured in the Sanders -Wyszeckl (1964) experiment.

20
experiment is still very accurate (r=0.983), as seen in figure 2, but the

model makes better sense physiologically. Although some predictions from the

author's ”Best-Fitting" model will be exhibited in later sections of this

paper, it is this ("Restricted") model that should be regarded as having some

potential for development into a general model of color vision.

6.1.3 Guth and Lodge, Original

Figure 3 shows the predictions of the last version of Guth's model (Guth and

Lodge, 1973) that did not contain adjustable constants. Guth's evolving model,

in its various forms, has been the most widely accepted model within the

vision/color academic community. It was singled out for discussion in CIE

Publication No. A1 (1978).

As seen in figure 3, this model falls to predict the Sanders-Wyszeckl data well

in three ways: (a) the predicted luminances are uniformly too low; (b) for

colors having low experimental equivalent luminances (in the 20-24 cd/m^ range),

the model predicts a virtually constant luminance just under 20 cd/m^; and

(c) the relationship between predicted and experimental values is highly

curvilinear.

It was to be expected that this model would not fit the Sanders-Wyszeckl data

as well as the author's models, since the brightness data that the Guth model

was Intended to fit was from a different experiment, conducted near threshold

levels. However, the gross discrepancy between data and model seen in figure 3

may seem rather surprising. Guth and his co-workers became aware some time ago

that their fixed model did not do well at predicting supra-threshold brightness

data, so that they proceeded to develop a generalized model with adjustable

21
PREDICTED

VALUES

(cd/mKKE)

EXPERItCNTAL VALIES (cd/miHfEJ

Figure 2, Correlation plot (scatter diagram) for restricted Howett model*

Predicted equivalent luminance is plotted against the equivalent

luminance measured in the Sanders-Wyszecki (1964) experiment*

22
40 MODEL: GUTH S LODGE. ORIGINAL

20 25 30 35

EXPERIMENTAL VALUES (cdM»S)

Figure 3. Correlation plot (scatter diagram) for original Guth and Lodge

model. Predicted equivalent luminance is plotted against the

equivalent luminance measured in the Sanders-Wyszecki (1964)

experiment.

23
parameters to allow for different conditions of luminance and chromatic

adaptation (Guth, Massof, and Benzschawel, 1980). Unfortunately, they have

to date provided only qualitative guidance as to how their six adjustable

parameters vary with the luminance level and chromaticity of the adapting

light, or other objectively measurable stimulus conditions.

6.1.4 Guth and Lodge, Adjusted

The difficulty (a) listed in section 6.1.3 can be compensated for by multiplying

all three equations In the model [Eqs. (1)] —or equivalently, the final

predictions [Eq. (2)] — by a suitable constant. The result of carrying through

this process Is shown In figure 4, the constant selected having the value that

gives the least-squares best fit to the data. Only the vertical scale of the

predictions Is affected by this constant adjustment, and the difficulties (b)

and (c) of section 6.1.3 are still present. The sum of squared errors of

prediction has been reduced from 2455 In figure 3 to 511 in figure 4. Since

linear (product-moment) correlation Is unaffected by applying a linear transformation —such as multiplication by a constant— to either set of values

being correlated, the value of r In figure 4 Is 0.816, the same as In figure 3.

It Is Interesting that the value of the constant factor that optimizes the sum

of the squared prediction errors is not what one might intuitively guess —i.e.,

the ratio of the mean of the experimental values to the mean value predicted by

the unadjusted model— but Is defined by a somewhat more complicated expression,

namely

96

k

I Bi Bi*

1-1

96

I 1-1

( 10 )

24
PREDICTED

VALUES

(cd/mxxPl

EXPERIMENTAL VALUES (cd/mxifE)

Figure 4, Correlation plot (scatter diagram) for adjusted Guth and Lodge

model. Predicted equivalent luminance is plotted against the

equivalent luminance measured in the Sanders-Wyszecki (1964)

experiment. The absolute scale of the Guth and Lodge predictions

(Fig. 3) was adjusted for optimum fit.

25
where is the ^th original predicted value, and B^ is the ^th experimental

value. In this instance, the numerical value of ^ turned out to be 1.22.

The expression (10) was derived as the least-squares solution to the general

problem of what constant one should multiply a given set of values by, in order

to best approximate a second set of values. Specifically, if {Bi} is any set

of II values, and |Bi*j any other set of n^ values, then the value of ^ that

minimizes

U - Z (kBj^-B^*)^ (10a)

i-1

is given by Eq. (10) [with the specific value 96 of Eq. (10) replaced by the

general jn ] . The derivation proceeds, as usual, by differentiating IJ of

Eq. (10a) with respect to k, and setting the resulting derivative equal to zero.

6.1.5 Thornton, Original

Thornton’s (1973) model, embodied in Eqs. (3)-(4), also fits the Sanders-Wyszeckl

data badly. As seen in figure 5, virtually all the predictions are overestimates,

and the relationship is also highly curvilinear. One gathers from his publications

that Thornton intended his model to fit "everyday" (supra-threshold) brightness

data; it would appear that he derived his formula to agree with some sort of

average of a number of heterochromatic brightness-matching studies that he cites.

6.1.6 Thornton, Adjusted

The overall scale of Thornton's model was adjusted by the best-fitting constant,

Ic, derived in accordance with Eq. (10), but figure 6 shows that the predictions

are still bad. The sum of the squared errors of prediction was reduced from

5749 in figure 5 to 1925 in figure 6 by multiplying the predictions by k-0.796.

26
PREDICTED

VALUES

(cd/mifxp)

MODEL: THORNTON, ORIGINAL

EXPERIMENTAL VALUES (cd/wxx2)

Figure 5. Correlation plot (scatter diagram) for original Thornton model.

Predicted equivalent luminance is plotted against the equivalent

luminance measured in the Sanders-Wyszecki ( 1964)

experiment.

27
Figure 6. Correlation plot (scatter diagram) for adjusted Thornton model.

Predicted equivalent luminance is plotted against the equivalent

luminance measured in the Sanders-Wyszecki (1964) experiment. The

absolute scale of the Thornton predictions (Fig. 5) was adjusted

for optimum fit.

28
The correlation coefficient is 0.616 in both cases. It is worth recalling that

the suras of squares associated with the author’s optimized models are 36 (figure

1) for the very best model and 44 (figure 2) for the restricted model. Thus,

the sum of squared errors of the author’s restricted model is better than that

associated with the original Guth and Thornton models by factors of 56 and 131,

respectively; and better than even the adjusted models by factors of 12 and 44,

respectively.

6.2 OPPONENT-LEVEL CHANNEL RESPONSES

The remainder of section 6 is devoted to showing the predictions of the various

models for several basic functions of color vision. A model that predicts

only brightness is valuable (especially in the lighting field), but a model

that also predicts other aspects of color perception with reasonable accuracy

is valued more highly by many vision specialists.

The figures shown as part of this subsection (6.2) are not primarily Intended

as predictions of empirical vision data, but represent the theoretical responses

of the opponent-colors channels implied by the various models. These functions

can in turn be used to predict experimental color data of various kinds. Guth

believes that the two opponent curves (T^ and ^) should be interpretable as

direct predictions of the perceived hues of the spectrum (red-green and blueyellow, respectively). If that assumption is accepted, then the figures of

this subsection can also be interpreted as predictions of one type of color

data.

29
6.2.1 Guth and Lodge

Guth’s channel responses [Eqs. (1)], shown in figure 7, are considered

reasonable in general form by most theorists. The luminance function, A, is

all-positive and very similar to the standard V(X) function. If these

functions are interpreted (as Guth does) as directly reflecting the perceived

colors of the pure (monochromatic) lights of the spectrum, this model fails to

reflect the fact that redness reappears at the shortwave end of the spectrum

(violet hue). There should be a small positive lobe of the red-green (dashed)

curve in the wavelength region from 400 up to perhaps 450 nm or higher.

It should be noted that most of the color-related functions that can be predicted

from the opponent -level channel responses do not depend on the absolute sizes

of the coefficients, but only on various ratios that can be formed from them.

Consequently, no further distinction will be made in this paper between the

original and adjusted models of Guth and Lodge (see section 6.1.4) and Thornton

(see section 6.1.6). Each subsection from here on will discuss the predictions

of four models only: Howett, Best-Fitting; Howett, Restricted; Guth and Lodge

(Original, but not so labeled); and Thornton (Original, but not so labeled).

6.2.2 Thornton

Thornton’s model [Eqs. (3)], depicted in figure 8, does predict the redness of

shortwave light. Its most unsatisfying feature is the double-humped yellowblue function (which virtually coincides with the red-green function above

about 540 nm). Note that Guth (figure 7) chose blue as the positive valence

of his blue-yellow function. The choice of sign is arbitrary and the dotted

curve in figure 7 can be mentally reflected in the abscissa axis for direct

comparison with the other models.

30
CHANNEL

RESPONSE

MODEL: GUTH & LODGE

HAVELEmTH M

Figure 7, Opponent-level channel responses for the Guth and Lodge model.

31
CHANNEL

RESPONSE

i,0

MODEL: THORNTON

Legend

LUMINANCE (BUCK-WHITE)

RED-GREEN

YELLOW-BLUE

-0.5

300.0 400.0 500.0 600.0

NAVELEN6TH (nm)

700.0 800.0

Figure 8. Opponent-level channel responses for the Thornton model.

32
6.2,3 Howett, Best-Fitting

The functions shown in figure 9 characterize the best unrestricted model optimized by the author [Eqs. (8)]. This model was derived entirely on the basis

of the Sanders and Wyszecki brightness data. There were no restrictions imposed

to make the model meaningful for other color predictions. Nevertheless, the

forms of these channel-response functions bear an encouraging resemblance to

the Guth and the Thornton functions, which were restricted to make sense with

respect to color predictions (especially Guth's). The luminance function

looks a good deal like V(X), as it should, but the shortwave negative lobe is

an unrealistic feature.

It should be mentioned that the Sanders and Wyszecki (1964) experiment involved

a large field of view, subtending 10° of visual angle. The chromaticity

coordinates given for the test colors in their paper were calculated on the

basis of the CIE 1964 supplementary 10° observer, rather than the more commonly

used 1931 2° observer. The a^^j coefficients that emerged from the optimizations

[Eqs, (8) and (9)] accordingly apply to tristimulus values within the 10°

system. However, the 10° and 2° observers, while differing significantly, do

not differ greatly. Therefore, the color functions predicted by the author's

two models were all constructed using the spectral tristimulus values of the

CIE 1931 2° observer, rather than the 10° observer to which the coefficients

actually apply. This was done to permit direct comparison with the predictions

of the Guth model and the Thornton model. Like the use of Guth's formulas for

predicting the various color functions (see section 6), the switch of colorimetric

systems provided a further handicap to the author's models, so that a reasonably

good showing relative to the other models might be taken as providing hope

33
MODEL: HOWETT, BEST FITTING

Figure 9. Opponent-level channel responses for the best-fitting Howett model.

34
that appropriate further adjustments would improve the predictions. Conducting

this study with this level of Imprecision also weighed against the application

of any statistical tests at this stage (see the end of section 5), although it

should be kept in mind that the prediction of equivalent luminances by the

author’s models was not subject to the handicaps that have been mentioned.

6.2.4 Howett, Restricted

The author’s model was re-optimized with the single restriction that the luminance curve be all-positive [Eqs. (9)]. Figure 10 now shows curves that make

about as much sense as Guth’s or Thornton’s curves. The reddish appearance of

the shortwave band of the spectrum is predicted (small positive red lobe).

However, there is a secondary negative (blue) lobe of the yellow-blue function

in the range of 590-700 nm, which does not correspond to hue perceptions in that

region. The same feature appeared in the wholly unrestricted model (Fig. 9),

and seems to be forced by the data.

Possibly, the interpretation that the opponent functions ^ and ^ directly

specify the perceived hues of the spectrum, in simple red-green/blue-yellow

terras, is an oversimplification. Perhaps another mathematical stage, of

other than purely linear form, must be Introduced to derive predictions of the

hues of the spectrum from the opponent-channel curves.

6.3 SATURATION OF THE SPECTRUM

The curves shown in this subsection predict the saturation (perceived degree of

difference from white) of spectrally pure lights. This function has been

measured experimentally many times, and the details of the curve vary from one

determination to another. The general form found is characterized by a rainlraura

35
MODEL: HOWETT, RESTRICTED

Figure 10, Opponent-level channel responses for the restricted Howett model.

36
saturation in the yellow, around 570-580 nm, and maximum saturation at both

extremes of the spectrum (violet /blue and red). All the models correctly

predict these gross features.

6.3.1 Guth and Lodge

Guth's model generates a saturation curve (figure 11) that probably resembles

the typical experimental determinations best. Note the bump near A80 or A90

nm.

The formula used by Guth for predicting the saturation of any color is

JtI+M

a+|t|+|d| •

The luminance or whiteness. A, in the denominator of Eq. (11) does not require

an absolute value sign since it Is never negative. The formula (11) has been

used for all the saturation predictions In this subsection.

Intuitively, formula (11) defines saturation as the fractional part of the

total absolute stimulation of the three opponent-level channels that arises

from the two chromatic (opponent) channels. Presumably, Guth tried several

different formulas for saturation and found that the expression (11) gave the

most realistic-looking function. However, the use of the simple sums of absolute

values appears more appropriate to Thornton's model than to Guth's. For the

general linear model. It would also seem appropriate to test the expression

( |t|p+|d|P)^/p , (12)

(ap+|t|p+|d1P)1/p

where the value of £ associated with the particular model Is used. The author

has not yet examined any predictive function other than formula (11).

37
MODEL: GUTH & LODGE

Figure 11. Saturation (perceived color strength) of the monochromatic lights

of the visible spectrum, as predicted by the Guth and Lodge model.

38
It is interesting that in the very latest model to which Guth has contributed

(Benzschawel and Guth, 1984), saturation is no longer predicted by the expression

(11), but rather by (12) with p«2, the characteristic value of Guth’s vector

model. This latest model, which is nonlinear, is intended primarily to predict

color differences, and it is not yet clear whether Guth regards It as a general

color-vision model.

6.3.2 Thornton

Thornton's spectral saturation function, shown in figure 12, exaggerates the

bump barely seen in Guth's curve (figure 11). Many functions of color vision

have minor bumps present, which often become considerably more conspicuous

under conditions of chromatic adaptation or adaptation to high levels of white

light. See Guth, Massof, and Benzschawel (1980) for data reflecting this

general phenomenon.

6.3.3 Howett, Best-Fitting

It is seen in figure 13 that the author's unrestricted model implies a saturation

function generally resembling Thornton's (figure 12). The high spike around

470 nm is unrealistic, and is presumably a consequence of the negative lobe of

the luminance function (figure 9).

6.3.4 Howett, Restricted

The author's restricted model generates a saturation function (figure 14) that

has lost the Incorrect spike (figure 13) and now bears considerable resemblance

to Thornton's curve (figure 12).

39
MODEL: THORNTON

Figure 12. Saturation (perceived color strength) of the monochromatic lights

of the visible spectrum, as predicted by the Thornton model.

40
Figure 13. Saturation (perceived color strength) of the monochromatic lights

of the visible spectrum, as predicted by the best-fitting Howett

model

.

41
Figure 14. Saturation (perceived color strength) of the monochromatic lights

of the visible spectrum, as predicted by the restricted Howett

model.

42
6.4 EQUIVALENT LUMINANCE OF THE SPECTRUM

When the overall spectral sensitivity of the eye is determined by flicker

photometry or certain other procedures, the result is the standard CIE V(X)

luminous efficiency function. When the criterion is changed to an appearance

of equal brightness (or certain other procedures), the sensitivity function

that results is similar to V(X), but broader, l.e., higher at the ends of

the spectrum. The latter curve also tends to be somewhat bumpier than V(X).

CIE Publication No. 41 (1978) provides a review of the two classes of

sensitivity-determination techniques.

In this subsection, the predictions by the four models of the latter

(brightness-based) function are examined. The author (Howett, in press)

refers to this function as "the equivalent luminance of the spectrum," or "the

equivalent luminous efficiency function," and symbolizes it as Vq(X). (No

standard terminology or symbol has yet been adopted by an international or

national organization.)

For each model, the equivalent luminance of the spectrum is calculated using

the spectral tristimulus form of Eq. (6), with the channel responses [Eqs. (5)]

and power, associated with that particular model. Explicitly, the equation

is

The function Vq(X) specifies the normalized (relative) equivalent luminance of

each wavelength in an equal-energy spectrum, just as V(X) corresponds to the

normalized luminance of each wavelength in an equal-energy spectrum. "Normalized"

(13)

43
in the preceding sentence (a more specific term than the more commonly used

"relative") means that all values along the curve were divided by the maximum

value, so that the maximum value of the normalized curve becomes unity.

6.4.1 Guth and Lodge

The equal-brightness "sensitivity" curve implied by the Guth model is shown as

the solid curve in figure 15. (This is a prediction of the curve that has

been referred to formally in this paper as the "equivalent luminance of the

spectrum," or the "equivalent luminous efficiency function.") Guth's curve

resembles experimental data more than the others do. The dashed curve is CIE

V(A). Both curves are normalized to unit maximum for comparison. Note

the slight bump in the neighborhood of 600 nm (here only an inflection).

6.4.2 Thornton

Thornton’s corresponding function (the solid curve in figure 16) has a more

conspicuous bump near 600 nm, and a very large bump —which is probably

unrealistic— around 450 nm.

6.4.3 Howett, Best-Fitting

The author's unrestricted model, shown as the solid curve in figure 17, predicts

an equivalent-luminance curve that strikingly resembles the Thornton prediction

(figure 16). Again, the large bump near 450 nm is unrealistic, and presumably

reflects the negative lobe in the luminance channel response (figure 9).

6.4.4 Howett, Restricted

The author's restricted model generates a curve (figure 18) in which the bump

at 450 nm is greatly suppressed, but still clearly present. The bumpiness near

and below 600 nm is also still evident. Note in figure 15 that even though

44
^ WwV • V I HAVELmm (na)

Figure 15. Equivalent (brightness-based) spectral luminous efficiency function

(solid curve), as predicted by the Guth and Lodge model. The dashed

curve is the standard (flicker-based) CIE luminous efficiency

function, both curves being normalized to unit maximum.

45
MODEL: THORNTON

Figure 16. Equivalent (brightness-based) spectral luminous efficiency function

(solid curve), as predicted by the Thornton model. The dashed

curve is the standard (flicker-based) CIE luminous efficiency

function, both curves being normalized to unit maximum.

46
MODEL: HOWETT, BEST-FITTING

MAVELEmW (rm)

Figure 17, Equivalent (brightness-based) spectral luminous efficiency function

(solid curve), as predicted by the best-fitting Howett model. The

dashed curve is the standard (flicker-based) CIE luminous efficiency

function, both curves being normalized to unit maximum.

47
ninii

iM

iii i

Diii l

nr

mninr m

i

iMlii

iiiiiiiiiiiiiiii

ll

iiiiiiiiiiiiiiiii

iil

iiiiiiiiiiiii

i

iiir

MODEL: HOWETT. RESTRICTED

V

400.0 450.0 500.0 550.0 600.0

mELENGTH (nm)

650.0 700.0

Figure 18. Equivalent (brightness-based) spectral luminous efficiency function

(solid curve), as predicted by the restricted Howett model. The

dashed curve is the standard (flicker-based) CIE luminous efficiency

function, both curves being normalized to unit maximum.

48
Guth's predicted curve is much smoother, slight inflections suggesting the

bump near 450 nm (as well as, more pronouncedly, that near 600 nm) are visible.

Published data resemble Guth's smoother curve more than the bumpier curve of

this figure.

6.5 WAVELENGTH DISCRIMINATION

The wavelength discrimination function is one of the classical curves of color

vision experimentation. As with the spectral saturation curve, there have been

many published determinations, with some variation in fine details but with a

few features that are always present. The curve plots the amount of wavelength

change that can just be visually detected (ordinate) against the starting

wavelength as abscissa. The threshold is usually taken to be the average of

the just noticeable wavelength shifts in the upward and downward directions from

the starting wavelength (except near the ends of the visible spectrum). All

four of the models being studied here predict wavelength discrimination curves

having the requisite gross features: two minima, in the blue and yellow portions of the spectrum; a maximum in the green region; and strong deterioration

of wavelength discrimination (high thresholds) toward both ends of the spectrum.

It is difficult to judge which model predicts the curve most closely resembling

empirical data, because the various experimental determinations differ so much

from each other in their details. No detailed, separate comments on figures 19-

22 will be made here, for that reason. It can be seen in figures 21 and 22

that both of the author's models generate slightly bumpier wavelength

discrimination curves than do the Guth and Thornton models (Figs. 19 and 20).

It is difficult to judge whether the various actual data curves do or do not

exhibit these minor irregularities.

49
THHESHOLD

HAVELENGTH

CHANGE

(RELATIVE)

Figure 19. Wavelength discrimination function, as predicted by the Guth and

Lodge model. The units on the ordinate axis are arbitrary.

50
THRESHOLD

WAVELENGTH

CHANGE

(RELATIVE!

250

^

200 —

150 —

100 —

50 —

du.

400

Figure 20.

MODEL: THORNTON

700

Wavelength discrimination function, as predicted by the Thornton

model. The units on the ordinate axis are arbitrary.

51
TmsmLD

WAVELENGTH

CHANGE

(RELATIVE)

Figure 21. Wavelength discrimination function, as predicted by the best fitting

Howett model. The units on the ordinate axis are arbitrary.

52
miESHOLD

tIAVELENGTH

CHANGE

(RELATIVE)

300

^ MODEL: HOWETT. RESTRICTED

250 —

200 —

iSO—

iOO —

50 —

OLl

400

Figure 22.

700

NAVELENGTH (na)

Wavelength discrimination function, as predicted by the restricted

Howett model. The units on the ordinate axis are arbitrary.

53
The mathematical basis for the predicted wavelength-discrimination curves is

the assumption, generalized from that made by Guth, that the total psychophysical

difference, AE, between any two colors Ai,Ti,Dj and A2 ,T 2 ,D 2 is given by

AE = [U1 -A2 P + |Tj-T

2 |P + |Dj-D

2 |P]^''P. (13a)

In order to isolate the ability to chromatically discriminate different

wavelengths from any brightness differences that might exist between the

wavelengths, we deal, again following Guth, with a constant-brightness spectrum.

It should be recalled that the functions a(X), t(X), and d(X), of the spectral

tristimulus form of Eq. (5), define the opponent-level channel responses for a

constant-radiance ("equal energy") spectrum. In order to obtain the channel

responses for a constant-brightness spectrum, we impose the condition that the

spectrum have constant equivalent luminance. The desired responses to yield

unit equivalent luminance are a(X)/Vq(X), t(X)/Vq(X), and d(X)/Vq(X), where

Vq(X) is the equivalent luminance of the spectrum, defined in Eq. (13).

Then, applying Eq. (13a), the total color difference between equally bright

lights of wavelengths X and X+AX is given by

AE

I Ip I Ip I Ip

|a(X+AX) a(X) I |t(X+AX) t(X) | |d(X+AX) d( X) | |Vq(X+AX) “ V^TXTI |Vq(X+AX) V^TxTi |Vq( X+AX) \

For each wavelength, X, in the spectrum, Eq. (13b) was applied to each model

(13b)

with AX = +1 nm and also AX = -1 nm; the two AE values for each model were

averaged; and the average was plotted against X to produce Figs. 19-22. This is

a direct generalization of the procedure specified by Guth and Lodge (1973).

6.6 NEUTRAL POINTS

As can be seen in Figs. 7-10, both the red-green and blue-yellow channels of

all opponent-colors models take on a value of zero at one or two wavelengths

in the interior (as opposed to the ends) of the visible spectrum. When the

54
red-green channel is at zero, neither red nor green is being signaled. Hence,

the color being perceived must be a pure yellow, a pure blue, or a neutral

(white, gray or black) —the only colors that stimulate no perception of either

red or green. Similarly, a zero value of the blue-yellow channel implies the

perception of a pure red, a pure green, or a neutral. Clearly, when both

opponent channels are at zero, the only possible perception is neutral; and,

conversely, any neutral (achromatic) color must fail to stimulate both opponent

channels.

The mathematical basis for specifying exactly where each opponent channel has

the value zero, for the general linear model, is provided by the last two of

Eqs. (5): i.e., the equations for the opponent responses ^ and ^ as a function

of _X, _Y, and Z, If we set the right side of either of these equations equal

to zero, we can translate into chromaticity terms (jc, and O by simply

dividing through by X + Y + Z. When 1-x-y is substituted for the resulting

expressions for the loci in the diagram along which and ^ are zero,

are respectively

T = 0: (a2i~a23)x + (a22“^23)y ^23 “ ^ (l^a)

D = 0: (a3i~a33)x + (a32~a33)y + a33 =0 . (14b)

Obviously, both of Eqs. (14) are of the form to represent straight lines in the

chromaticity diagram. Moreover, the intersection of the straight lines having

the equations (14a) and (14b) is the (unique) point corresponding to a neutral

perception.

55
An explicit formula for the neutral point of the general model can be obtained

by solving Eqs. (14) simultaneously, the result being

X = (a22 a33-a23 332 )/g (15a)

y = (^23 ^31"^21 ®33)/g , (15b)

where

g

= a3i(a23-a22) + a32(^21"^23) ^33 (^22“^2l) • (15c)

The only situation in which the neutral point falls to exist is when the two

null lines of the opponent channels (Eqs. (14)] are parallel. Algebraically,

this condition is equivalent to the quantity g of Eq. (15c) being zero. The

non-intersection of the null lines is highly "unnatural," and it seems unlikely

that any model incorporating this feature would make color predictions that

resemble data adequately.

In the process of deriving Eqs. (14) and (15), we have incidentally provided a

geometric proof of an important theorem. We see in Eq. (6) that ^ (equivalent

luminance) is equal to A (luminance according to the model) if, and only if,

both and ^ are zero. We have shown that ^ is zero if and only if the chromaticity point of the color lies on the line represented by Eq. (14a) and ^ is

zero if and only if the point lies on the line represented by Eq. (14b).

Hence, B = A —or the ratio B/A = 1 — if and only if the point is the intersection of these two null lines. Since two lines Intersect in a single point if

they intersect at all, we have established that either (a) the value of B/A is

unity at one specific point (the neutral point) and is greater than unity

everywhere else [because the two terms other than A in Eq. (6) are both nonnegative, so that their sum is positive everywhere except at the neutral point];

56
or, (b) [in the case where the null lines are parallel] there is no point for

which B/A - 1, and the ratio is greater than unity everywhere.

In representations of experimentally determined constant-B/A (or B/Y or B/L)

contours in the literature, there appears to remain some doubt as to whether

the locus upon which B/A=l is a single point, a closed curve, or an entire

area. However, the preceding arguments show that whatever the actual facts may

be, the prediction of the general linear model is that the locus is a single

point.

It should be noted that in the preceding derivation, the model has been taken at

face value, and the model's own formula for luminance (A) has been used, rather

than the currently defined measure of luminance (jf). For a model having a

definition of A not identical to Y, B/Y will not in general be equal to unity

at the neutral point, although B/A must be unity. Such a model is asserting

that the CIE 1931 y(X) function [» 1924 V(X)j is not a perfectly correct

representation of the spectral response of the eye's luminance channel. It

behooves us to accept all the predictions of a model as is, especially since an

error in y(X) is widely acknowledged actually to exist (Judd, 1951). Later, in

the discussion of the constant-B/A ovals of the general model, it will be seen

that there is a compelling computational simplicity associated with the use of

A rather than Y for luminance (when the two are not the same).

6.6.1 Guth and Lodge

Figure 23 shows that in the Guth and Lodge model, the line (dashed) of no

redness-greenness extends from approximately the shortwave end of the spectrum

locus to a region that can reasonably be described as yellow. As explained in

57
6UTH S LODGE

Figure 23. Loci of zero red-green (dashed) and zero blue-yellow (dotted), as implied by the Guth and Lodge model. The intersection of the two

lines is the white (neutral, achromatic) point of the model.

58
the opening paragraph of section 6.6, this line must contain the chromaticity

points of colors perceived as pure blue, white, and pure yellow. Unfortunately,

the shortwave end of the spectrum is not blue but violet; i.e., the perceived

color contains a definite admixture of redness. Thus, a well-known defect of

the Guth model is pointed up by the figure.

The line (dotted) of no blueness-yellowness has both ends in not quite the

right places. The stimulus producing a perception of pure red does indeed lie

on the purple boundary, rather than at the longwave end of the spectrum. The

point where the dotted line in Fig. 23 hits the purple boundary is, however,

too far toward the violet end, and corresponds to a purplish red with a definite

component of perceived blue. Similarly, the other end of the dotted line cannot

be described as the pure green wavelength of the spectrum; it corresponds to a

clearly bluish green.

Finally, the neutral point of the Guth and Lodge model —the Intersection of

the dashed and dotted lines — is also a little on the bluish side, perceptually.

In the later version of the model (Guth, Massof and Benzschawel, 1980), the

definitions of the opponent channels were modified in a way that resulted in a

neutral point closer to what is usually considered pure white. As explained

earlier, the present author has not attempted to explore the properties of the

latter model, since the choice of adaptation constants changes most of the

implications of the model, including the neutral point.

6.6.2 Thornton

Figure 24 reveals a serious defect of the Thornton model. The line (dashed) of

no redness-greenness extends from the blue-green part of the spectrum to the

59
MODEL: THORNTON

Figure 24, Loci of zero red-green (dashed) and zero blue-yellow (dotted), as implied by the Thornton model. The intersection of the two lines

is the white (neutral, achromatic) point of the model, but in this case the point has the chromaticity of a spectrally pure yellow.

60
yellow; and the line (dotted) of no blueness-yellowness extends from the purple

to the yellow. The result is that the "neutral" point of the Thornton model

Instead of being in the central region perceived as white, is actually a

spectrally pure yellow, of wavelength very close to 581 nm. (Careful computation shows that the point corresponds to an imaginary color just beyond the

spectrum locus, but negligibly so.)

6.6.3 Howett, Best-Fitting

The dashed line in Fig. 25, which should join the pure blue part of the spectrum

locus to the pure yellow part, does fall in approximately the right location.

The dotted line, which should join the pure red and pure green points, is

clearly in the wrong place. It joins oranglsh red to a slightly bluish green.

Considering that this model was derived entirely on the basis of brightness

matches, its performance on this test is by no means disgraceful, compared to

that of the Guth and Lodge or Thornton models.

6.6.4 Howett, Restricted

Comparison of Figs. 25 and 26 shows that the author's model restricted to give

an all-positive luminance-channel function is a bit worse than the very bestfitting model with regard to the opponent-channel null loci. Both lines are

rotated slightly in the wrong direction. That is, in the restricted model,

the blue point is further toward violet, and the green point is further toward

blue. The red point —the right end of the dotted line— is still a quite

oranglsh red. The neutral points of both models are within the region usually

identified as white, toward the yellowish side, with the restricted model's

white point being a bit less yellow.

61
Figure 25. Loci of zero red—green (dashed) and zero blue-yellow (dotted), as implied by the best-fitting Howett model. The intersection of the

two lines is the white (neutral, achromatic) point of the model.

62
Figure 26. Loci of zero red-green (dashed) and zero blue-yellow (dotted), as implied by the restricted Howett model. The intersection of the

two lines is the white (neutral, achromatic) point of the model.

63
6.7 THE PRIMARIES

In a traditional, three-component (receptor-stage) color system, without opponent

channels, the chromaticity of one of the three primary colors of the system can

be located by setting both of the other two primaries equal to zero. (This

leads to pure "stimulation" by only the third primary). This procedure is

precisely what was done [Eqs. (14)] in the previous subsection to locate the

neutral point of an opponent-colors system. Two of the three channel responses

(namely the two chromatic, opponent channels) were set equal to zero. It is

natural to wonder about the meaning of the points located by setting equal to

zero both members of each of the two other possible channel pairs. Thus, colors

that can be identified by analogy as the "primaries" of an oppponent-colors

model can be derived from the three conditions T=D=0,A=T=0, and

A = D = 0. The author is not currently able to give an interpretation of what

meaning is to be attached to the three "primary" colors of an opponent-colors

system.

As we saw with Eqs. (14), setting two of the channel responses of a linear

opponent-colors model equal to zero corresponds to intersecting two straight

lines on the chromaticity diagram, thereby determining a unique point. The

meaning of a zero value for one of the opponent (chromatic) channels is clear,

but what is the interpretation of A = 0? Since A is the luminance channel,

the line A = 0 is the locus of colors that the model assigns zero luminosity;

in short, it is the "lightless line" or alychne of traditional, three-component

chromaticity diagrams. The alychne of the CIE 1931 diagram is the line

y

= 0, which of course lies wholly outside of the region of real colors; all

real colors have positive luminosity. Since every point on the line A = 0

64
therefore corresponds to an imaginary color, we know that the intersection of

any other line with A = 0 determines a chromaticity point corresponding to an

imaginary color. In other words, we know in advance that two of the three

"primaries" of an opponent-colors system are always imaginary. The third

"primary," determined by intersecting the lines T = 0 and D = 0, can be either

real or imaginary, depending on the definitions of the opponent channels

associated with the particular model. Ordinarily, since the point corresponding

to the conditions T = D = 0 is the neutral point of the model (section 6.6),

it should be expected that the third primary —the one corresponding to pure

stimulation of only the A channel-should be real and should correspond to a

white color. This reasoning may help to make clearer why the A channel is

spoken of in different contexts as being alternatively the luminance or the

white (achromatic, neutral) channel.

6.7.1 Guth and Lodge

The alychne of the Guth and Lodge model is close to y =0, as seen in Fig. 27.

This reflects the fact that Guth's A channel is defined [Eq. (1)] as very close

to _Y (or y). Any reasonable model that explicitly identified its A channel

as corresponding to luminance would be expected to define A as nearly or

actually equal to Y, since y is known to be close to the correct weighting

function for calculating luminance. This will be confirmed graphically in the

other figures of this subsection, as well as in the accompanying tables.

It is important to recall that in deriving the author's best-fitting model, no

assumptions at all were made about the nature of the three channels, beyond

that there were three channels and that the spectral response of each of them

was expressible as a linear combination of x, y, and z. Thus, it is significant

65
Figure 27, The opponent-level primaries (stars) of the Guth and Lodge model.

These are the points for which two of the channel responses A, T,

and D are zero. The upper two lines of the triangle (T“0 and D*0) are the same as those in Fig, 23; the bottom line is A“0 (the

alychne of the model).

66
that what emerged from the optimization was one almost entirely positive channel

whose response clearly resembled y; and two opponent channels that made reasonable

sense as red-green and blue-yellow responses. This outcome suggests that

accounting for brightness-matching data may require a model that considerably

resembles contemporary opponent-color theories.

Even the author's restricted model was subject only to the single requirement

that one of the channels have all-positive response. There was no further

requirement that this channel have a spectral response at all resembling y,

in the sense that any positive coefficients of x and z could have emerged.

Nevertheless, the data forced the model to have its all-positive channel

essentially identical to y [see Eqs. (9)], and again the other two channels

took on opponent forms that could not unreasonably be identified as red-green

and (perhaps somewhat less clearly) blue-yellow. Again, this result can be

construed as suggesting the possible reality of the basic opponent-colors

notion.

Moreover, it should not be forgotten that in Guth's earliest work, which

involved the factor analysis of a matrix of data representing the additivities

at threshold of lights of different wavelengths, he too found that the data

implied the existence of a channel with a response resembling y, and two

opponent channels. Thus, the opponent-colors model, in broad outline, seems

to be more than an abstraction that can be made to account for certain vision

data; instead, more than one kind of vision data seems to directly imply the

essential truth of that model.

Returning now to the specific characteristics of the Guth and Lodge model, we

observe that the point (indicated by a star) at the lower left corner of the

67
triangle of primaries In Fig. 27 Is the Intersection of the lines corresponding

to A “ 0 and (see Fig. 23) T “ 0. The point therefore determines the primary

corresponding to pure stimulation of the £ or blue-yellow channel. This

point is just outside the shortwave end of the spectrum locus, in the same

area where the tritanopic convergence (sometimes called ”co-punctal") point

(the missing blue primary of the tritanope) is traditionally located.

The lower right point of the primary triangle, which corresponds to the redgreen primary, is located very nearly on an extension of the longwave end of

the spectrum locus —the line z = 0 or x + y

= 1. It is in a region where some

authors locate the convergence (co-punctal) point (missing primary) of

deuteranopes. Its color would therefore be ordinarily identified as green

(not red).

Table 1 provides accurate numerical information, for the Guth and Lodge model,

on both the equations of the null lines and the coordinates of the primaries.

One primary of any of these models is always (at least nominally) white, but

it is not obvious what criterion should be used to decide whether an imaginary

red-green primary of an opponent-colors system should be labeled either red or

green, particularly when it is located directly on the alychne of the system.

Consequently, the practice followed in Table 1, and in the analogous tables for

the other three models, was to use the non-commital terms "red-green primary"

and "blue-yellow primary".

6.7.2 Thornton

The primaries of Thornton's model, as revealed in Fig. 28, are quite different

from those of the Guth and Lodge model (Fig. 27). What is presumably supposed

to be a blue primary (lower left) is far outside the spectrum locus, and is in

68
Table 1. Guth and Lodge Model; Null Lines and Primaries of Opponent-Level

Channels

A - 0; y

•= 0.010593X - 0.010593

Wavelength^ x y

Outside region of real colors.

T - 0: y

- 2.016702X - 0.348643

Wavelength^ x L

360.739

570.371

0.17550

0.44666

0.00529

0.55213

D - 0; y

» -0.340909X + 0.340909

Wavelength^ 2L y

491.470 0.03862 0.32774

C491.470 0.51773 0.16441

Primary Conditions x y

Blue-yellow

Red-green

White

A - T » 0

A » D - 0

T - D = 0

0.168510 -0.008808

1.000000 0.000000

0.292479 0.241200

® The wavelengths (nm) and corresponding chromaticitles are the intersection

points of the null lines with the spectrum locus or purple boundary. The

letter C denotes "complementary" with respect to the neutral (white) point

of this model » and indicates a point on the purple line.

69
J.O MODEL: THORNTON

-f.O

i.5 —J

-i.5 -i.O -0.5 0.0 0.5 i.O

A'

Figure 28. The opponent-level primaries (stars) of the Thornton model. These

are the points for which two of the channel responses A, T, and D

are zero. The upper two lines of the triangle (T"0 and D"0) are

the same as those in Fig. 24; the bottom line—here coinciding with

the x-axis — is A-0 (the alychne of the model).

70
more of a blue-green direction. Neither this nor the other chromatic primary

(lower right) is in a location that is ordinarily thought of as a convergence

(co-punctal) point for any type of dichromatic vision. The direction of the

red-green primary is here actually toward the purple. As has been previously

noted (in 6.6.2), the "white" primary is in fact a spectral yellow in Thornton's

model.

The numerical specifications of Thornton's null lines and primaries are given

in Table 2. The null lines T = 0 and D = 0 have very nearly —but not quite

converged in the neighborhood of 581 nm on the spectrum locus; the intersection

point is barely beyond the locus.

6.7.3 Howett, Best-Fitting

In Fig. 29, the red-green primary is very far off to the right. More

disturbingly, the blue primary (lower left) is a real color, in contradiction

of the assertion made in the introductory remarks of subsection 6.7. The dashed

line just above the x-axls (y=0) is the alychne, yet it intersects a corner

area of the region of real colors! The consequence is that some real colors

(violets) are assigned zero and negative luminosities (the latter being the

colors within the "horseshoe" that lie below the alychne). This physically

Impossible situation is a consequence of the negative lobe of the A function

for this model (Fig. 9), and is the reason the model was cast aside.

Table 3 summarizes the numerical specifications of the null lines and primaries

for this model.

71
Table 2. Thornton Model: Null Lines and Primaries of Opponent -Level

Channels

A * 0: y

“ 0

Wavelength^ x Outside region of real colors.

T = 0: y

= 0.301000X + 0.323805

Wavelength^ x Z

491.786

580.998

0.03718

0.51906

0.33499

0.48004

D = 0: y

= 2.976110X - 1.064979

Wavelength^ x Z

581.008 0.51912 0.47998

C581.008 0.39371 0.10674

Primary Conditions X y

Blue-yellow A = T = 0 -1.075765 0.000000

Red-green A = D = 0 0.357842 0.000000

White T = D = 0 0.519150 0.480069

^ The wavelengths (nm) and corresponding chromaticitles are the intersection

points of the null lines with the spectrum locus or purple boundary. The

letter C denotes "complementary" with respect to the neutral (white) point

of this model, and indicates a point on the purple line.

72
MODEL: HOWETT. BEST- FITTING

Figure 29, The opponent-level primaries (stars) of the best-fitting Howett

model. One point (the red-green primary) is off the figure to the

right. These are the points for which two of the channel responses

A, T, and D are zero. The upper two lines of the triangle (T-O and

D=0) are the same as those in Fig, 25; the bottom line is A“0 (the

alychne of the model).

73
Table 3 Howett Best-Fitting Model: Null Lines and Primaries of OpponentLevel Channels

A - 0: y

- -0.054A75X + 0.108950

Wavelength^ X y

477.055 0.10258 0.10336

C477.055 0.35668 0.08952

; y

= 1.589946X

Wavelength^

- 0.153770

X y

468.258 • 0.12846 0.05048

570.135 0.44501 0.55377

: y

= -0.190283X + 0.510433

Wavelength^ X y

498.799 0.01102 0.50834

595.193 0.60392 0.39552

Primary Conditions x y

Blue-yellow

Red-green

White

A - T - 0

A - D - 0

T » D “ 0

0.159765 0.100247

2.956245 -0.052092

0.373100 0.439438

® The wavelengths (nm) and corresponding chromaticities are the intersection

points of the null lines with the spectrum locus or purple boundary. The

letter C denotes "complementary" with respect to the neutral (white) point - of this model, and indicates a point on the purple line.

74
6.7.4 Howett, Restricted

The blue primary In Fig. 30 is in about the same position as it has in the Guth

and Lodge model (Fig. 27). The white primary is reasonably situated, in a

slightly yellowish part of the white area. The most striking feature is the

location of the red-green primary very far off to the left. The author is

currently unable to Interpret this result or correlate it with any particular

features of the predictions of this model.

Table 4 gives the numerical data for the null lines and primaries of this

model.

6.8 CONTOURS OF CONSTANT B/A

In the past, authors who have wanted to exhibit contours of constant B/A

(equivalent luminance over luminance) on the chromatidty diagram, for their own

model, have calculated B/A at each point of a fairly finely spaced grid, and

then sketched in each contour by interpolating smoothly by eye along a path

passing near points with similar B/A values. This is the same method used with

empirical data, such as elevation contours on topographical maps, or Isobars and

isotherms on weather maps. However, it is possible to derive an explicit

equation for the locus B/A k (k > 1), for the general linear model.

Applying Eqs. (5) and (6), we can write

bP - |ajjX+aj2Y+a|3Z|P + |a2iX+a22Y+a23Z|P + |a3jX+a32Y+a33Z|P. (16)

We want to set B/A equal to the specific value i.e.,

B - kA, (17)

or kPB vPaP » k A . (IgN

75
MODEL: HOWETT, RESTRICTED

Figure 30, The opponent-level primaries (stars) of the restricted Howett model.

One point (the red-green primary) is off the figure to the left.

These are the points for which two of the channel responses A, T,

and D are zero. The upper two lines of the triangle (T“0 and D»0) are the same as those in Fig, 26; the bottom line —here virtually

coinciding with the x-axls —is A-0 (the alychne of the model).

76
Table 4. Howett Restricted Model: Null Lines and Primaries of OpponentLevel Channels

A “ 0: y

= -0.001521x

Wavelength^ x

Outside region of real colors.

T = 0: y

= 1.921231X - 0.265231

Wavelength^ x Z

455.571 0.15025 0.02343

568.379 0.43266 0.56601

D = 0: y

= 0.016667x + 0.404167

Wavelength^ x L

^9^.672 0.02478 0.40458

591.795 0.58548 0.41392

Primary Conditions x Z

Blue-yellow

Red-green

White

A = T = 0 0.137943 -0.000210

A = D = 0 -22.222600 0.033790

T = D = 0 0.351470 0.410025

^ The wavelengths (nm) and corresponding chromatlcities are the intersection

points of the null lines with the spectrum locus or purple boundary. The

letter C denotes "complementary” with respect to the neutral (white) point

of this model, and indicates a point on the purple line.

77
The right sides of Eqs. (16) and (18) are both equal to BP, and hence are equal

to each other:

1<PaP = |ajiX+aj2Y+a^3Z|P + |a2iX+a22Y+a23ZlP + | a3jX+a32Y+a33Z|P. (19)

Since we know from Eq. (5) that

A= |aiiX+ai2Y+ai3Z| , (20)

and therefore that

aP = |ajjX+aj2Y+aj3ZlP, (21)

we can replace AP in Eq, (19) by its equivalent given by Eq, (21), The result

is

kP|a^^X+aj2Y+aj3Z|P = ja^j X+aj2Y+aj3Z|P + | a2iX+a22Y+a23Z| P +

|a3iX+a32Y+a33ZlP, (22)

or

(kP-1) |ajjX+aj2Y+aj3Z|P = 1 a2 jX+a22Y+a23 z] P + |a3jX+a32Y+a33Z|P. (23)

Equation (23) can be converted from tristiraulus values to chromaticity coordinates by dividing the entire equation through by |X+Y+Z|P, The result is

(kP-1) |ajjX+aj2y+aj3z|P = |a2j>H-a22y+a23z|P + | a3j3cfa325H-a33Z | P, (24)

Finally, substituting for ^ its equivalent,

z = 1-x-y, (25)

and rearranging terms, we arrive at the final, explicit equation

(kP-1) I (ajj-aj3)x+(aj2~ai3)y+a23 jP = | (a2i-a23)x+(a22“a23)y+a23

+ 1 (a3j-a33)x+(a32“a33)>H-a33 jP, (26)

Because of the power, jp, as well as the absolute values, Eq, (26) cannot be

explicitly solved for ^ as a function of x,

6,8,1 The Special Case p=2 (Including the Guth Model)

In order to use Eq. (26) to generate the B/A«k contours for a particular model.

78
it would appear that some sort of iterative numerical procedure must be employed

in order to obtain the value of _y corresponding to a given x. However, there

is one special case in which Eq. (26) reduces to a recognizable form: namely,

p=2, as in the models of Guth and of Ingling, Squaring an absolute value

permits the dropping of the absolute value signs, so in this case Eq. (26)

reduces to an ordinary second degree equation in x and Such an equation

represents a conic section. Because the B/A ovals are closed curves in the

neighborhood of the white point of the model (B/A = 1), we know that Eq. (26)

must represent an ellipse in that region. It is conceivable that at some

distance from the white point (i.e., for large k values), the discriminant of

Eq. (26) changes sign, and the locus becomes a different category of conic,

such as a hyperbola or parabola, but this possibility has not been examined

algebraically by the author.

6.8.2 The Special Case p=l (Including the Thornton Model)

One other case, p=l , is slightly less obvious, but also leads to loci of an

easily specified form. If we could disregard the absolute value signs in

Eq. (26), with p=l , the result would be a collection of terms of three types:

a constant times x; a constant times y^; and simply a constant. Combining

these terms accordingly leads to the equation of a straight line in the chromatlclty diagram.

However, we are not free to disregard the absolute value signs in Eq. (26).

Each of the two expressions within the absolute value signs on the right side

can be either positive or negative (or zero) for real colors. Fortunately,

this leads to only four distinct cases: the signs of these expressions can be

79
+ or - No other possibilities exist. (The zero case is not

significant, and can be lumped with either + or - or both.) By assigning

explicit signs to the expressions inside, we can then drop the absolute values,

because we know that for any quantity |s|=s if s>0; and | s

| =-s if s^O.

The term inside the absolute value signs on the left side of Eq. (26) is always

positive for real colors, for any model in which the alychne does not intersect

the region of real colors; which is to say, for any reasonable model. Hence,

those absolute value signs can also be dropped. The result, when all the

terms are combined, is, in each of the four cases, the equation of a straight

line. Points lying on any one of these four lines satisfy Eq. (26). Thus, we

have established that for p=l , the loci B/A=k are formed from four straight

lines; i.e., they are quadrilaterals rather than smoothly curved ovals.

If a model in which some real colors are permitted to have negative luminosity

is to be considered, then two more possibilities are introduced: the expression

within the absolute value signs on the left side of Eq. (26) can also be either

positive or negative. Combining this dichotomy with the four sign possibilities

arising on the right side of the equation leads to a total of eight cases

(2x2x2), each of which leads to the equation of a different straight line. In

this situation, the loci of constant B/A are octagons, or possibly some more

complex eight-sided figures, such as separated pairs of quadrilaterals.

6.8.3 The General Case (Including the Howett Model)

So far, except when p=l , it does not appear as if Eq. (26) can be used easily

to trace out constant-B/A contours with precision. However, it turns out,

perhaps surprisingly, that Eq. (26) can be converted into a form that easily

permits direct, polnt-by-polnt plotting through an explicit formula, without

80
resort to Iteration. The unique status of the white point (primary) of a

model—the sole point for which B/A=l —suggests that setting up a polar

coordinate system centered there might lead to some kind of simplification.

Indeed, converting to a polar form allows the separation of variables, and the

derivation of an explicit formula for the radius as a function of the angle.

The formula can be written more compactly by Introducing abbreviations for

some of the key quantities in Eq. (26). Let

Rp = (kP-l)l/P (27)

^11 = ail " ai3, 6 i 2 = ai2 - (28)

<S2l = a2i - a23, ^22 = ^22 “ a23 (29)

<531 = a3i - a33, 632 = a32 - a33. (30)

Then the formula is

Rp(^llXo+5l2yo+ai3)

r’ ^ (31)

{ I 62 iCos 0 *+

622 sln 0 '

I P + 1 52icos0’+622sia0' | P}l/P - Rp(6ncos0*+6i2sln0') ,

where Xo,yo are the chromaticity coordinates of the white point of the model;

and r', 0 ’ are polar coordinates in the shifted system centered at XQ,yQ.

Equation (31) follows from Eq. (26) by applying to Eq. (26) the Inverse translation equations

X = x’ + xq, y = y' + yo; ( 32 )

and then converting to polar coordinates relative to the new origin, in

accordance with the standard equations

x’ = r'cos0’, y' = r’sin0’. (33)

The rest of the derivation is algebraic manipulation. Account is taken of the

fact that the white point lies on both of the lines T=0 and D=0, and hence

81
Xo,yo satisfies both Eq. (14a) and Eq. (14b). (This reduces several terms

to zero.) The final complete separability of r' seems to be valid for any

color for which A, the luminance of the model, is positive; i.e., for any real

color, in a reasonable model. The author has not explored algebraically the

validity of Eq. (31) for imaginary colors with negative luminosity.

Equation (31) has been used to generate the contours in Figs. 31-34, for the

four models being investigated. (Being entirely general, it applies to the

cases p=l and p=2, without any special considerations, as well as to all other

2_ values.) The primed coordinates of Eq. (31) can be converted back into unprimed

CIE chromaticity coordinates for plotting, by applying Eq. (33) to r’,9’ and

then applying Eq. (32) to x*,y'.

Although Eq. (31) is a valid formula, certain rather unexpected difficulties

arise in its actual application. The source of these difficulties is not yet

understood. In essence, false sections of some of the curves are generated, a

phenomenon somewhat analogous to the appearance of physically meaningless roots

of some polynomial equations. These "ghost" curve segments tend to appear

above or below the region of real colors, but sometimes encroach upon that

region.

The problem appears to be associated with the fractional form of Eq. (31), as

opposed to the original, non-f ractional Eq. (26). The denominator of Eq. (31)

can be positive, negative, or zero. When it is near zero, Eq. (31) calls for

an exceedingly large radius, and trouble seems to stem at least partly from

this source.

There is a relatively simple solution to the problem. In plotting the oval

82
B/A“k, examine each point generated by formula (31). Using the numerical

versions of Eqs. (5) and (6) for the particular model, the actual value of B/A

can be calculated for the proposed point. If the radius r' is negative, or if

the actual value of B/A for the proposed point differs excessively from the

value Ic supposedly characterizing all points on the curve, reject the point and

simply plot no point for that value of 9*. In drawing Figs, 31-34 (by computer),

any point was rejected if its actual B/A ratio differed from the nominal value

by more than 5x10“^ (in either direction).

6.8.4 Guth and Lodge

As anticipated in section 6.8.1, this model, with p=2, yields smooth curves

which are conic sections (Fig. 31). The most striking feature of Fig. 31 is

the seemingly rather abrupt transition, somewhere around a B/A value of 1.0035,

from closed ellipses near the white point, to what appear to be hyperbolas,

with very straight asymptotic tails. The possibility of such a shift was

pointed out in section 6.8.1, but the values of the appropriate constants

needed to confirm this effect have not been calculated, either for the general

linear model with p=2, or for the Guth and Lodge model in particular.

6.8.5 Thornton

As anticipated in section 6.8.2, the Thornton model, with p=l, yields constantB/A "curves" which are straight-sided quadrilaterals (Fig, 32). Thornton

himself (1973) was aware of this implication of his model, but chose to grossly

round off the corners in presenting the corresponding figure in his 1973 paper.

Presumably, this was because he believed that nature tends not to have sharp

corners, except at the quantum level. There is a very slight rounding of some

of the corners in Fig. 32, but this effect is an artifact of the computer

83
Figure 31. The loci of constant B/A (equivalent luminance over luminance)

as predicted by the Guth and Lodge model. Because p"2 for this

model, the curves are all conic sections (see text). At least some of the curves (those for low B/A values) are clearly ellipses

(the asymmetrical clipping of some of the tips being an artifact of

the plotting routine) , whereas the curves for higher B/A appear to

be hyperbolas. Fifteen ovals are shown. The B/A values, from

Inside (small) to outside (large) are; 1.0003, 1.0009, 1.002,

1.003, 1.0034, 1.01, 1.04, 1.06, 1.09, 1.11, 1.13, 1.16, 1.2, 1.3,

and 1.4. The B/A value at 360 nm, the extreme shortwave end of

the spectrum, is only 1.92 for this model. The plus sign (center

of smallest oval) denotes the white point of the model, the only

point at which B/A-1.

84
MODEL: THORNTON

Figure 32, The loci of constant B/A (equivalent luminance over luminance)

as predicted by the Thornton model. Because p=l for this model,

the loci are straight-sided quadrilaterals (see text), with sharp

corners. Ten "ovals" are shown. The B/A values, from inside

(small) to outside (large) are: 1,08, 1,17, 1,31, 1,5, 1,75, 2,0,

2,7, 4,0, 7,0, and 35,0, The B/A value at 360 nm, the extreme

shortwave end of the spectrum, is a probably unrealistic 67.6

for this model. The plus sign denotes the "white" point of the

model, the only point at which B/A=l,

85
plotting routine. In actuality, the intersections are sharp points.

Visual alignment in Fig. 32 suggests that all of the upper and lower corner

points of the quadrilaterals lie on a single straight line, and that all of the

left and right corners similarly lie on a single line. This appearance has not

yet been confirmed by algebraic analysis. If there are indeed these two

principal axes in the space, it is an interesting question as to what visual

significance these directions have within the model. Comparison with Figs. 24

and 28 suggests that the alignment axes in Fig. 32 may correspond to the lines

T=0 and D=0 within the Thornton model. This has not yet been confirmed

algebraically.

6.8.6 Howett, Best-Fitting

The most novel aspect of Fig. 33 is the shape of the constant-B/A loci. They

are four-sided figures with very sharp corners (cusps), and with sides that are

concave outward. Comparison of Figs. 31 to 33 suggests the possibility that

all models with powers greater than unity may have loci that are convex outward;

all models with powers less than unity may have loci that are concave outward;

and we do know that all models with a power exactly equal to unity have straightsided loci (neither convex nor concave).

As with the Thornton quadrilaterals of Fig. 32, the "diamonds” of Fig. 33 seem

to have their sharp corners aligned along the T=0 and D=0 lines of the model

(see Figs. 25 and 29).

There is one extraordinary feature of Fig. 33. The lowest locus shown is close

to the alychne (A=0) of the model (see Fig. 25). Because of division by zero,

the constant-B/A locus coinciding with the alychne has a B/A value of infinity.

86
Figure 33. The loci of constant B/A (equivalent luminance over luminance)

as predicted by the best-fitting Howett model. The equations for

the "ovals" can be obtained by substituting the values from

Eqs. (8)-(8a) into Eq. (26), or Eq. (31). Thirteen "ovals" are shown. The B/A values, from inside (small) to outside (large)

are; 1.14, 1.23, 1.30, 1.36, 1.41, 1.46, 1.49, 1.52, 1.55,

1.75, 2.0, 3.0, and 4.0. A little below the line for

B/A=4.0 lies the alychne (A=0) , along which this model has

infinite values of B/A. Below that, B/A becomes finite again,

and reaches an absolute value of 3.17 at 360 nm, the extreme

shortwave end of the spectrum. (The issue of whether the values

below the alychne should conventionally be regarded as positive

or negative is discussed in the text.) The plus sign denotes

the white point of the model, the only point at which B/A“l.

87
Below that, the B/A ratio drops again to finite values. The sign of B/A below

the alychne is a matter of convention. Because of the absolute values in its

definition [Eq. (6)], ^ is always positive. Thus, from this point of view, when

A goes negative, the ratio B/A also goes negative. As the alychne is crossed

moving downward, the ratio switches from positive infinity to negative infinity,

and thereafter rises (algebraically) to a modest -3.17 at the shortwave corner

of the spectrum.

An alternative viewpoint is that it is reasonable always to give JB the sign of

_A; i.e., if the luminance (A) of a light is negative, then it seems appropriate

for its equivalent luminance (JB) to also be negative. By that interpretation,

the ratio B/A is always positive, and, after reaching infinity on the alychne,

drops back down to successively lower positive values as ^ continues to decrease.

The B/A ratio at the shortwave end of the spectrum is, by this reasoning, best

regarded as +3.17. The author tends to favor this latter convention, because

it preserves the previously derived theorem that the ratio B/A can never be less

than unity. If the first convention is adopted instead, then B/A values can only

be greater than +1 or less than -1 , so that a gap in the permissible range of

values then exists.

Of course, this entire discussion normally applies only to imaginary colors,

lying outside the region of the chromaticity diagram that is of practical interest. In the case of the best-fitting Howett model, with which Fig. 33 deals,

the critical crossover region Involves physically realizable colors that can

actually be observed in an experiment. This completely unrealistic behavior

is still another aspect of why this model is not considered worth pursuing, in

comparison with the restricted Howett model.

88
6.8.7 Howett, Restricted

Except for the bizarre behavior caused by the alychne running through the region

of real colors in Fig. 33, the restricted Howett model illustrated in Fig. 34

has very similar features. Cusped diamonds may seem even harder to believe

than Thornton's quadrilaterals, but the excellent fit of this model to the

Sanders-Wyszecki data suggests that only a model that implies some such extreme

shape —possibly slightly rounded—could agree well with that body of data.

6.9 ADDITIVITY

6.9.1 The Significance of Nonadditivity

Luminance is calculated, except for a possible constant factor, by weighting

the spectral radiance of a light, wavelength by wavelength, by the y(X) [=V(X)]

function, and summing these products across the entire visible spectrum. As a

consequence of this quantitative definition, luminance is additive; that is,

the luminance of the mixture of any two (or more) lights is equal to the sum of

the luminances of the component lights.

Is equivalent luminance similarly additive? Since the definition of equivalent

luminance is the luminance of an equally bright white light, an operational

test of this question would run as follows: given, for example, a red light

and a green light, match the red light in brightness by adjusting the luminance

of a reference white light; similarly match the green light; and finally, match

the mixture of the red and green lights. Is the luminance of the white light

that matches the red-green mixture in brightness equal to the sum of the white

luminances that match the red and green lights separately in brightness? As Is

now well known, the answer to the question is no. For most observers, the white

luminance that matches the red-green mixture in brightness is significantly

89
MODEL: HOWETT, RESTRICTED

Figure 34, The loci of constant B/A (equivalent luminance over luminance)

as predicted by the restricted Howett model. The equations for

the "ovals" can be obtained by substituting the values from

Eqs. (9)-(9a) into Eq. (26), or Eq. (31). Fifteen "ovals" are shown. The B/A values, from inside (small) to outside (large)

are: 1.06, 1.12, 1.20, 1.25, 1.30, 1.33, 1.37, 1.40, 1.43,

1.45, 1.6, 2.0, 3.0, 4.0, and 6,0. The B/A value at 360 nm, the

extreme shortwave end of the spectrum, is 15.9 for this model,

a high but not ridiculous value. The plus sign denotes the

white point of the model, the only point at which B/A“l.

90
less than the sum of the white luminances that brightness-match the red and

green lights separately. Guth refers to this result as subadditivity. Where

the equivalent luminance of a mixture exceeds the sum of the equivalent luminances

of the component lights, Guth applies the term superadditivity. [These terms

are widely used in the literature, and will be used here, but some authors

(Kaiser and Wyszeckl, 1978) prefer the expressions "additivity failure of the

cancellation type," and "additivity failure of the enhancement type," respectively.]

It is now well established that superadditivity does occur for some combinations

of lights, although subadditivity is more common and more quantitatively extreme

(Kaiser and Wyszecki, 1978; Burns et al., 1982).

The frequent discrepancy between equality of luminance and equality of brightness,

for lights of different chromaticities , provides one of the two great challenges

to color-vision models; the other is provided by the nonadditivity phenomenon.

In developing his model, Guth confronted both of these challenges. His assumption

that the brightness-perception center of the visual system is fed not only by

the luminance channel, but also by the chromatic channels, accounts in a

qualitative way for the Imperfect correlation between brightness and luminance.

His further assumption (not originated by him, of course) that the chromatic

channels of the eye are opponent in nature allowed him to account qualitatively

for nonadditivity. It is worth observing that a nonopponent (Young-Helmholtz)

color vision model that assumes that one or more of the chromatic channels

feeds into the brightness-perception center, together with a composite luminance

channel, would also predict some discrepancy between luminance and brightness,

but would entail perfect additivity. Thus, the two problems are closely related,

but not logically equivalent.

91
6.9.2 Vectorial Considerations

Figure 35 is a sketch of the well-known principle of vector addition, shown in

two dimensions. In the case shown, with a large angle between Q and R, the

length of the sum vector Q+R is considerably less than the sum of the lengths

of Q and R. When the angle between Q and R is small, the length of Q+R is only

slightly less than the sum of the lengths of Q and R. When the angle is zero

i.e., when Q and R are in the same direction—then the length of the sum vector

is exactly equal to the sum of the lengths of the component vectors.

The review of the basic concepts of Guth's vector model given in section 1.1

indicates that, for him, Q and R would represent colors and Q+R would represent

the mixture color formed by superimposing the lights represented by Q and R.

Guth's space is three-dimensional, the axes being the quantities A, T^, and ^

of Eqs. (1) and (2). The lengths of the vectors are proportional to the

equivalent luminances (not the brightnesses) of the corresponding colors. It

was indicated in the preceding subsection (6.9.1) that a simple index of

additivity, as Illustrated in Fig. 35, is the length of Q+R minus the sum of

the lengths of Q and R. When this quantity is negative, subadditivity is

Indicated; when it is positive, superaddltlvlty is indicated; and when it is

zero, exact additivity is Indicated. Because Guth's space has a euclidean

metric [Eq. (2)], there is no arrangement of the vectors Q and R (Fig. 35) that

can ever yield superadditivity.

6.9.3 Definition of a Generalized Nonadditivity Index

In terms of the generalized linear model represented by Eqs. (5) and (6), let

us denote the length (norm) of a vector C by |C|^P^, the superscript (p) indicating

the power in Eq. (6). Then we can define a generalized additivity—or really

92
Figure 35. The principle of vector addition. Q and R (solid_line) are vectors radiating from the origin. We translate R parallel

to itself, ^ntil the tail of R (dashed) is touching the head

(arrow) of Q. Then the vector directed from the^ o^rigln to

the head of the translated R is the sum vector Q+R.

93
nonadditivity—index, applying for any value of as follows. For any colors

and Cj , and the mixed color , the index is defined as

P^P\q,Cj) = 100

(34)

By dividing the crude index mentioned earlier (6.9,2) by the sum of the vector

norms, the ^ index of Eq. (34) becomes independent of the absolute sizes of the

vectors; i.e., of the absolute luminances of the colors. The factor 100 converts

the dimensionless index to a percentage, rather than a fractional basis.

Thus, a ^ value of -10 indicates that the norm of the sum vector (the equivalent

luminance of the mixed color) falls short by 10 percent of equaling the sum of

the separate vector norms (the sum of the equivalent luminances of the separate

colors). This corresponds to what we can now describe as 10 percent subadditivity.

Similarly, a ^ index of +5 denotes 5 percent superadditivity, A ^ value of 0

denotes perfect additivity. For the sake of precision of thought, ^should not

be described as an additivity index, but as a nonaddltlvity index; the bigger

the absolute value of the greater the degree of nonadditivity. For purposes

of recalling the significance of the sign of the number, ^ can be thought of

even more specifically as an index of percent superadditivity; but, in any

case, it seems natural to associate "plus" with "super" and "minus" with "sub."

6.9.4 The Relationship of the Power, to Nonadditivity

It was pointed out in section 6.9,2 that when p=2 in Eq. (6) (the euclidean

metric), a prediction of superadditivity is impossible. This is a significant

failure of the Guth model and of the Ingling model. Is there any value of p

94
for which superadditivity can be predicted? In an attempt to find out how £

Influences the possible values of the jP index, a Monte Carlo computer simulation

was run. For simplicity, two-dimensional vectors were studied. The two

components of each of two vectors were chosen by a random number generator that

produced a uniform distribution of values ranging from -2 to +2. (This range

was chosen in case values greater than unity in absolute value behave differently

than values less than unity in absolute value.) The degree of superadditivity

resulting from the addition of these randomly selected vectors was calculated

in accordance with Eq. (34). Several hundred pairs of such randomly chosen

vectors were examined for each value of £. used.

The results of the Monte Carlo runs were quite unequivocal. For all values

exceeding unity, by even a very small increment, superadditivity never occurs.

In other words, for p>l , subadditivity always occurs (except for the vanishingly

rare cases when the two vectors happen to lie along the same direction, or one

of them is the zero vector, in which cases additivity occurs for any 2.)* For p=l

,

subadditivity is still the most common result, but perfect additivity occurs

with significant frequency (about 25 percent) for vectors not lying along the

same direction. It is only for jo exactly equal to 1 that this phenomenon is

observed. Superadditivity does not occur for p=l. For values less than unity,

by even a very small decrement, subadditivity continues to be the most common

condition, but superadditivity occurs about 25 to 30 percent of the time.

Although it was obvious that the percentages might well vary with the dimensionality of the vector space, it seemed highly likely that the present author's

models, with their powers in the neighborhood of 0.82, would be able to predict

superadditivity in some cases. Whether these cases would correspond to the

mixtures of real colors remained to be determined.

95
6.9.5 Actual Predictions; Pairs of Monochromatic Lights

Using the restricted model embodied in Eqs. (9)-(9a), additivity calculations

were carried out, using the ^ index of Eq. (34), for mixtures of monochromatic

lights. A complete matrix of combinations of wavelengths at 10-nm Intervals

over the visible spectrum was computed, for various luminance ratios of wavelength

1 (X^) to wavelength 2 (X

2 ). The result for a 10:1 luminance ratio is shown in

Fig. 36. The model of Eqs. (9)-(9a) can indeed predict superadditivity, and

subadditivity as well. The surface depicted in the figure was somewhat smoothed

by the plotting routine, but the principal features shown are accurate. The

greatest degree of superadditivity is predicted by the model for long wavelengths

combined with short wavelengths in a 10:1 ratio of the former to the latter

(X 1 /X 2 ). The greatest degree of subadditivity is predicted for middle wavelengths

of \i combined with either short or long wavelengths of X 2

* Nearly perfect

additivity is predicted for short wavelengths of Xj , regardless of the wavelength

X 2

. The particular luminance ratio of 10:1 was chosen for Fig. 36 to illustrate

marked degrees of both sub- and superadditivity. Much larger and much smaller

ratios of Xj to X 2 yield predictions clustering more closely around the P=0

(additivity) plane.

6.9.6 Actual Predictions; Lines of Constant Dominant Wavelength

In an important paper combining theory and experiment. Burns et al. (1982)

examined color mixture along lines of constant dominant wavelength. Such a

line is traced out on the chromatlcity diagram by combining a white with a

monochromatic light in varying proportions, ranging from 0 (pure white light)

to 1 (pure spectral light). The departures from additivity for the various

mixtures along these lines were measured experimentally. Surprisingly, it

96
Figure 36, Predictions by the restricted Howett model [Eqs, (9)-(9a)] of

the superadditivity index, ^ [Eq. (34)], for mixtures of two

monochromatic lights. The plot shows ^ as a function of two

wavelengths, presented in a mixture in the luminance ratio of

10:1 of X 2 to X

2

»

97
was found that for some dominant wavelengths, there was superadditivity at low

purities (near white), with a switch over to subadditivity at high purities

(near the spectrum locus). Of course, in such cases, there was one intermediate

purity at which exact additivity held.

In their mathematical analysis of various types of models. Burns et al. (1982)

appeared to conclude that no simple linear model, and not even some nonlinear

models that they considered, could account for such results. However, careful

inspection of the paper reveals that the generalized linear model [Eqs. (5)-(6)]

for values of 2^ less than unity was not examined by those authors. This omission

was in all likelihood not an oversight; within the class of Minkowski Lp metrics

[i.e., Eq. (6)], the range p<l is not considered to represent valid distance

functions. Lp for any p<l is sometimes referred to as a "pseudometric." A

true metric must satisfy certain basic mathematical conditions, and one of them

is the triangle inequality. This condition requires that no side of any triangle

can have a length greater than the sum of the lengths of the other two sides.

In the light of all the preceding discussion of additivity and nonadditivity,

within section 6.9, it should be clear that violation of the triangle inequality

is precisely what is required if a quasi-metric is to be able to predict

superadditivity. In fact, superadditivity nothing more nor less than

violation of the triangle inequality. Any true metric, which satisfies the

triangle inequality, can never predict superadditivity.

The use of a value less than unity in the model proposed in this report [Eqs.

(9)-(9a)] means that the space corresponding to the model can be expected to

exhibit strange or even bizarre behavior. It must be realized, however, that

the facts of brightness perception—the data that the model is trying to predict

98
—are themselves strange or bizarre, and so far no predictions have been

generated from the model that are absurd on their face.

The final additivity test that was carried out was to determine whether the

restricted model of this paper [Eqs. (9)-(9a)] can predict the finding by Burns

et al. that a crossover from one direction of nonadditivity to the other can

occur along a line of constant dominant wavelength. Perhaps surprisingly, it

turns out that it can. Figure 37 summarizes the results of these computations.

What is plotted is the nonadditivity or superadditivity index, of Eq. (34),

as a function of dominant wavelength and colorimetric purity (the latter being

the ratio of the monochromatic luminance to the sum of the white and monochromatic

luminances). Each slice parallel to the purity axis shows the predictions for

a particular dominant wavelength. Each slice represents a two-dimensional

graph showing the superadditivity index ^ as a function of purity, for that

dominant wavelength.

As is shown by the frontmost slice, for a wavelength of 700 nm, the prediction

is that there is subadditivity at low purity values, additivity at a colorimetric

purity of about 0.2, and then awfeadditivity for purities beyond 0.2. Perfect

additivity is predicted for the specific purities of 0 and 1.0, for all

wavelengths, because in those two cases there is only one light in the mixture,

and the general linear model [Eqs. (5)-(6)] for any ^ always predicts exact

additivity when any light is added to itself or to nothing.

For short dominant wavelengths, particularly strong superadditivity is predicted

at low purity values, and here there is no crossover; there is superadditivity

for all but the terminal purities. Thus the qualitative pattern found by Burns

et al. (1982) —crossover of the type of nonadditivity at some dominant wavelengths

99
Figure 37. Predictions by the restricted Howett model [Eqs. (9)-(9a)] of

the superadditivity index, ^ [Eq. (34)], for mixtures of white

with monochromatic lights of different wavelengths. Each vertical

section parallel to the purity axis shows the variation of predicted ^ with colorimetric purity (i.e. , mixtures of the white

and monochromatic lights in various luminance proportions). The

chromaticity of the white used for the predictions was that of

the actual white source used by Burns et al. (1982).

100
and nonadditivity of a single sign at other wavelengths —is predicted by the

proposed model. Unfortunately, the details of the predictions differ sharply

from the Burns et al. experimental data. For example, the predicted crossovers

shown in Fig. 37 are from subadditivity at low purities to superadditivity at

high purities, whereas the Burns et al. data exhibit the reverse direction of

crossover. In their paper. Burns et al. used an indicator of nonadditivity

different from the _P index [Eq. (34)] used here. Their index, as well as was

calculated, and the two measures agree in their qualitative pattern. Thus the

discrepancies between the predictions illustrated in Fig. 37 and the Burns et

al. data must be due to a cause other than the different nonadditivity indexes

used.

Kaiser and Wyszecki (1978) showed that additivity information can be derived

from the Sanders-Wyszecki (1964) experimental data used as the basis for the

model optimizations described in the present report. They presented only

sketchy results, enough to show that both subadditivity and superadditivity are

implicitly present in the Sanders-Wyszecki data. Since the model proposed in

this report [Eqs. (9)-(9a)] predicts the Sanders-Wyszecki data so closely (see

Fig. 2), it is reasonable to assume that a full additivity analysis of the

Sanders-Wyszecki data would lead to a pattern closely resembling Fig. 37. If

that is true, then we have to confront the question of why the Sanders-Wyszecki

and Burns et al. data sets disagree with each other so much. Such disagreement

seems to prevail between nearly any two studies in the literature. Kaiser and

Wyszecki (1978) seem to feel that a substantial part of the problem is the use

of different groups of observers in the various experiments. That may be a

factor, but it is also true that virtually every experiment in brightness

perception is done with a different set of viewing conditions, experimental

101
procedures, and instructions to the observers. A systematic study of all these

variables to pin down the sources of the disturbingly high inter-experiment

variability seems to be a pressing need.

It has been demonstrated here that a linear model with a power less than unity

can generate additivity predictions with the kinds of complex properties that

characterize actual experiments. There seems reason to hope that if the general

linear model [Eqs. (5)-(6)] were to be re-optimized to fit the Burns et al.

data, a good fit might be possible, including reasonably detailed prediction of

the complex additivity findings. One expectation that seems close to a certainty

is that the value of 2. the best-fitting Eq. (6) would be significantly less

than unity.

7. SUMMARY AND CONCLUSIONS

The present author believes that averaging heterochromatic brightness-matching

data from experiments carried out under different viewing conditions tends to

obscure —rather than sharpen—the quantitative relationships underlying the

behavioral data. As a result, this study represents an attempt to account

mathematically for the data of a single, extensive experiment (Sanders and

Wyszecki, 1964).

The essence of the approach was to formulate a generalized linear model of the

opponent-colors type, and then optimize the constants of the model for best fit

to the brightness-matching data. On the assumption that the optimization

succeeded in locating the true global minimum of the sum of the squared prediction

errors (and, unfortunately, it is not certain that this assumption was

unequivocally fulfilled), it follows that the resultant model must necessarily

fit the data better than any other linear, opponent-colors model can. A crucial

102
question is: how good is this best fit? That is, does the model predict the

data well enough, in an absolute sense, that it would be considered a practical

tool for calculating brightness matches among lights of different colors? The

answer is yes: the predictions of the derived model correlate over 0.98 with

the matching data.

The next crucial question is: how well would this formula generalize to

predicting other sets of brightness-matching data? This question has not been

answered here. It is certain that this one formula cannot accurately predict

the data of all the experiments in the literature, since the various studies

disagree sharply with each other in a quantitative—but not qualitative—sense.

Such viewing parameters as the sizes of the fields being compared in brightness,

the separation between them, the luminance level of the lights, the presentation

time if the viewing is flashed, and the chromaticity and luminance of the

background, all are expected to influence equal-brightness judgments between

lights of different colors. Ultimately, a formula for predicting equivalent

luminance (the luminance of an equally bright reference white light) should

incorporate explicit adjustment terms for each of these factors. Thus, a

great deal of experimental, mathematical, and theoretical work clearly remains

to be done before such a highly refined formula can be derived.

The formula for equivalent luminance proposed by the present author [Eqs. (5),

(6), (9), and (9a)] is recommended for testing in "real-life" situations. It

is of particularly simple form: it represents a linear opponent-colors model

with a combination rule (metric) different from previous proposals. Although

evidence is accumulating that some sort of nonlinearity needs to be introduced

in an equivalent-luminance formula in order to account for certain color-vision

103
data, it is of interest to explore how well the best possible linear formula

can do. That is why an optimization procedure was used in the present study.

Even if the new formula were to be regarded as "mere curve-fitting," it would

be desirable to see how useful it proves to be in practical lighting work

involving lights with significant coloration. If it proves to be more realistic

than other formulas currently available, it can be used until something better

comes along. However, the present study has shown that the model of color

vision implied by the formula is capable of predicting at least the gross forms

of some of the classical color-vision functions, so that something more than

"mere curve-fitting" may be represented by the model. It is also an encouraging

sign that the channel responses that resulted from a completely unrestricted

optimization of the generalized linear model [Eqs. (5)-(6)] turned out to

consist of a nearly all-positive channel resembling the y(X) function, plus two

opponent channels corresponding more or less to red-green and blue-yellow

responses.

Moreover, when a new optimization was done with one channel restricted to being

all-positive everywhere within the visible spectrum, that channel turned out to

be essentially equal to y, and again the other channels were opponent in form

and more or less identifiable as red-green and blue-yellow. Thus, heterochromatic brightness-matching data seem to directly imply that processes

resembling opponent-colors theory are actually operative in the eye. Some

further exploration of linear versions of that theory may still prove informative, although nonlinear versions may take over as further data are published.

The present author's formulation of the generalized linear model involves

defining equivalent luminance as the p^th root of the sum of the p^th powers

104
of the three channel responses. For Guth and for Ingling, p=2, and for

Thornton, p=l. The optimizations performed as part of the present analysis

show that a power in the neighborhood of 0.82 is necessary to account for at

least the particular body of heterochromatic brightness data that was studied.

This is one of the most distinctive findings of the present study, and its

Implications would seem to deserve further exploration. The analysis of the

known nonadditivity of equivalent luminances carried out in this report has

already shown that no linear model with a value exceeding unity can predict

superadditivity at all; and that the author's model, with its power less than

unity, can predict the sort of complex pattern of mixed sub- and superadditivity

that characterizes real brightness-matching data.

7.1 SUGGGESTIONS FOR FURTHER WORK

A great deal of additional work in the general subject area covered by this

report is desirable in order to clarify the many unsettled issues. It is clear

that such further exploration needs to include both experimental and computational

investigations. Three examples of each type are described in the following

subsections

.

7.1.1 Some Proposed Experimental Studies

7 . 1 . 1 . 1 Is Chromatic Brightness Visually Effective ?

Strongly colored light (other than yellow) certainly looks brighter than whiter

light, but can it help us to see better? An experiment to answer this question

needs to concern itself with suprathreshold vision, since the eye's functioning

at spatial or temporal threshold appears to be controlled by luminance, rather

than brightness. If chromatic brightness does make a contribution to suprathreshold visual performance, a potential arises for saving energy by using

105
lower illuminances of somewhat chromatic light, without loss of visual

function.

7 . 1 . 1 . 2 How Much Coloration in a Light Source Is Tolerable?

An important limitation on the use of strongly chromatic lights in everyday

work environments is imposed by people’s frequently observed intolerance for

prolonged exposure to such lights. Presumably, the upset that people often

feel when forced to work under substantially chromatic lights is associated

with the progressive degradation of the color rendering of familiar objects as

a light source departs increasingly from white. It would be very useful to

know how saturated a light source, in every hue direction, is acceptable to the

average room occupant during long-term exposure.

7. 1.1.3 The Effect of Viewing Conditions on Equivalent Luminance

It is of crucial importance to explain the quantitative disagreement among the

various experimental studies of heterochromatic brightness matching in the

literature. Part of the variation almost surely arose because of the use of

different viewing conditions by the various investigators. What is needed is a

parametric investigation of the effects on brightness perception of separate

and simultaneous variation of a small number of principal properties of the

stimulus fields. A first experiment might examine the influences of the size

and separation of the fields being compared, as well as the luminance level.

Later studies could concern themselves with the relevant properties of the

background field, particularly its chromaticity and luminance. A key goal is

the inclusion of explicit functions of the more critical of these parameters

as modifiers of the constants in color-vision models.

106
7.1.2

Some Proposed Computational Studies

7. 1.2.1 Individual Observer Variation

Some of the published experiments on heterochromatic brightness matching include

data for the Individual observers. If an efficient optimization routine can be

discovered, it would be of great interest to optimize the constants of the

general linear model for each observer separately. It would then be possible

to note which parameters of the model tend to remain relatively constant over

different observers, and which ones tend to vary the most. The correlations

among the parameters could also be examined. If enough correlation exists, an

attempt could be made to characterize each observer by a small number of

constants from which all the constants of the observer's best-fitting model

could be generated.

7. 1.2.2 Studies of Nonlinear Models

Various types of nonlinear models need to be investigated. It is important to

determine how much better various color-vision functions can be predicted by

those nonlinear models than by the best linear models. The statistics of

nonlinear prediction might be applied in this connection, for significance

testing. A determination should also be made as to what color-vision phenomena

are correctly predicted by the nonlinear models, but not by any linear model.

7. 1.2.3 Further Predictions of Color-Vision Functions by The

Optimized Linear Model of This Report

Determination of how well the model embodied in Eqs. (9)-(9a) predicts colorvision functions not considered in this report should be informative. In

addition, attempts should be made to improve the fit of some of the functions

that have been considered here, by revising the equations postulated as connectln>*

the model to the various color-vision functions.

107
8. REFERENCES

Benzschawel, T. ; and Guth, S. L. (198A), "ATDN; toward a uniform color space,"

Color Research and Application , Vol. 9, No. 3 (Fall), pp. 133-141.

Burns, S. A.; Smith, V. C. ; Pokorny, J. ; and Eisner, A. E. (1982). "Brightness

of equal-luminance lights," Journal of the Optical Society of America , Vol.

72, No. 9 (Sept.), pp. 1225-1231.

CIE (1970). International Lighting Vocabulary , 3rd Edition, CIE Publication No.

17, Central Bureau of the CIE, Paris.

CIE (1978). Light as a True Visual Quantity; Principles of Measurement , CIE

Publication No. 41, Central Bureau of the CIE, Paris.

Guth, S. L. ; and Lodge, H. R. (1973). "Heterochromatic additivity, foveal

spectral sensitivity, and a new color model," Journal of the Optical Society

of America , Vol. 63, No. 4 (April), pp. 450-462.

Guth, S. L. ; Massof, R. W. ; and Benzschawel, T. (1980). "Vector model for normal

and dichromatic color vision," Journal of the Optical Society of America , Vol. 70, No. 2 (Feb.), pp. 197-212.

Howett, G. L. (in press). "The coming redefinition of photometry." Scheduled

for publication in Journal of the Illuminating Engineering Society, Vol. 15,

No. 3, April 1986.

Ingling, C. R. , Jr. (1977). "The spectral sensitivity of the opponent-color

channels," Vision Research , Vol. 17, No. 9, pp. 1083-1089.

Ingling, C. R. , Jr.; and Tsou, B. H.-P. (1977). "Orthogonal combination of the

three visual channels," Vision Research , Vol. 17, No. 9, pp. 1075-1082.

Judd, D. B. [unattributed] (1951). "Report of Secretariat of Technical Committee

No. 7, Colorimetry and Artificial Daylight," CIE Proceedings of Twelfth

Session, Stockholm, 1951 , Vol. 1, Central Bureau of the CIE, New York, p. 7/11.

Kaiser, P. K. ; and Wyszecki, G. (1978). "Additivity failures in heterochromatic

brightness matching," Color Research and Application, Vol. 3, No. 4 (Winter),

pp. 177-182.

Sanders, C. L. ; and Wyszecki, G. (1964). "Correlate for brightness in terms of

CIE color matching data," CIE Compte Rendu, Fifteenth Session, Vienna, 1963 , Vol. B, CIE Publication No. 11, Central Bureau of the CIE, Paris, pp. 221-230.

Thornton, W. A. (1973). "A system of photometry and colorimetry based directly

on visual response," Journal of the Illuminating Engineering Society , Vol. 3,

No. 1, pp. 99-111.

Vos, J. J. (1978). "Colorimetric and photometric properties of a 2° fundamental

observer," Color Research and Application , Vol. 3, No. 3 (Fall), pp. 125-128.

108
APPENDIX: BASIC COLORIMETRIC QUANTITIES AND NOTATION

CAPITAL LETTERS

Formal colorimetric systems are based on the fundamental notion that light of

any color can be matched by a mixture of three fixed lights called primaries.

The amounts of the three primaries needed to match any color are called the

tristimulus values of that color.

For each of the primary lights, a particular luminance is defined, in effect,

as unit amount of that primary. (The definition is usually indirect, and

numerical luminance values for the units are rarely seen, as such.)

Tristimulus values are specified relative to these unit quantities of each

primary. Thus, each tristimulus value is a ratio of one luminance to another,

and is dimensionless.

Tristimulus values are conventionally denoted by capital letters corresponding

to the designations of the primaries within the system. In the CIE 1931 standard

colorimetric system, the tristimulus values are designated _X, Y, and Zy

corresponding to the amounts of the particular red, green, and blue primaries

of that system, respectively. (The underlining here only indicates Italic

letters; see the Note on Notation immediately preceding section 1 of this report.)

In specifying the tristimulus values of a group of samples, the absolute sizes

of the tristimulus values of each sample have significance relative to the

sizes of the tristimulus values of the other samples. However, the overall

absolute scale of tristimulus values is arbitrary. In the case of patches of

light (the type of stimuli referred to in this paper), a common convention Is

to set the If value of each stimulus equal to its luminance, in some convenient

109
units such as footlamberts or nits (cd/m^). The and ^ values are adjusted

proportionally. Thus, the relative luminances of any two samples can be compared

by simply noting their respective Y values.

In opponent-colors systems, the quantities analogous to tristimulus values are

also denoted by capital letters. Accordingly, following Guth (Guth and Lodge,

1973), this paper uses the notation A, and ^ for the amounts of stimulation

of the luminance, red-green, and blue-yellow channels, respectively.

LOWER-CASE LETTERS

When the issue of luminance is secondary, and only color (in the popular sense)

is of interest, it is only the proportions of the tristimulus values to each

other that are relevant, and not their absolute size. Hence, the quantities

^ x+y+z ’ ^ X+Y+Z ’ ^ X+Y+Z

are defined, and referred to as chromaticity coordinates . From the definitions

of 21» L given in Eq. (Al), it is easy to see that: (a) the chromaticity

coordinates sum to unity; and (b) the tristimulus values are proportional to

the chromaticity coordinates, the proportionality constant being Xl-Y+Z. If, as

is common, stimuli are specified by x» chromaticity coordinates

and the luminance of the light —then the ^ coordinate is calculated as 1-x-y;

and the X and Z tristimulus values are retrieved by multiplying x and

respectively, by the ratio Y/y. Because of property (a), only two of the

chromaticity coordinates are independent. Therefore, the standard chromaticity

diagram represents the basic luminance-independent aspects of color in the form

of a plot of against x» with no reference to £.

no
The CIE 1931 x,y chromaticity diagram has one conspicuous failing: its spacing

is not uniform. A uniform [or uniform-chromaticity-scale (UCS)] diagram is one

having the property that the distance between any two chromaticity points on

the diagram is proportional (at least approximately) to the amount of difference

perceived between the colors corresponding to the points. In 1960, the CIE

recommended, for use as an approximately uniform chromaticity diagram, an

alternate set of coordinates designated _u and _v. The u,v diagram is described

mathematically as a projective transformation of the x,y diagram, which means

that straight lines on the x,y diagram are transformed into straight lines on

the u,v diagram, and vice versa. One of the most important properties of a

chromaticity diagram is that all colors that can be formed by mixing two fixed

colored lights in different proportions have chromaticlties that lie along the

straight line segment in the diagram joining the chromaticlties of the two fixed

colors. Consequently, the u,v diagram, despite its different spacing, represents

exactly the same facts about color mixture as does the x,y diagram.

Although this report does not make any use of the opponent-colors analogues of

chromaticity coordinates, if such quantities had been used, they would have

been denoted by the lower-case letters a_, and

LOWER CASE LETTERS WITH OVERBARS

The first experimental step in the development of a system of colorimetry is to

determine the amounts of the three primaries needed to match any monochromatic

(single-wavelength) light within the visible spectrum. (In practice, these

data are measured at intervals of several nanometers—usually 5 or 10 nm—and

then interpolated to 1-nm Intervals.) These are the tristimulus values of the

spectrum. The particular spectrum that is used is usually an "equal-energy"

111
spectrum: one produced by the emission of equal amounts of power (as measured

by radiance, for example) within each small, constant wavelength band across

the visible spectrum.

These spectral tristimulus values, which are referred to in older literature as

"distribution coefficients," are denoted by lower-case letters with overbars,

as: x( X) , y(X), and z(X). The (X) [Greek lambda] indicates functional

dependence on wavelength, and is redundant, since lower-case overbarred

letters are used only in connection with the spectral tristimulus values.

Therefore, the simple symbols x, y, z are frequently used without explicit

inclusion of X.

One of the most important properties of tristimulus values is that they are

additive: that is, the tristimulus values of a compound stimulus obtained by

superimposing several lights is equal to the sum of the tristimulus values of

the separate, component lights. Consequently, the tristimulus values of any

stimulus that has been analyzed into its component wavelengths—that is, whose

spectral power distribution is known

can be obtained by adding together the

tristimulus values of the spectrum, weighted in each wavelength band by the

amount of energy (radiance) in that band in the spectrum of the stimulus. In

short, the spectral tristimulus-value functions x( X) , y(X), and z(X) constitute standard weighting functions which, when used to weight the spectral

power distribution of any light, yield, after totalling across the entire

spectrum, the tristimulus values of the light. That is how tristimulus

values are usually actually calculated.

Still another way of describing the same arithmetic is to say that x(X),

112
y(X) and z(X) represent the spectral sensitivities of three "receptor

mechanisms" on which light is shone. The total response of the x mechanism

when any light is "absorbed" in it is the _X tristimulus value; and similarly

for the other two "mechanisms." In the case of CIE understood

that the x, y, and z "mechanisms" are formalisms only, and do not represent,

for example, the absorptions of the three types of color receptors in the

actual human eye. Nevertheless, it can be shown that x(X), y(X), and z(X)

must be linear combinations of the spectral sensitivities (absorptances ) of

the actual eye mechanisms. The spectral tristimulus functions of any other

colorimetric system which contains the same Information as the CIE system must

likewise be linear combinations of the photoreceptor sensitivity functions of

the eye. The conclusion is that all (infinitely many) CIE-equivalent colorimetric

systems can be expressed as linear transformations of each other. Accordingly,

one may work in any system convenient to the task at hand, and then easily

convert later to some fixed reference system, such as the CIE 1931 system.

Because of the relationship, described above, of the CIE system to the actual

color receptor mechanisms of the eye, it is assumed that any two light stimuli

having equal tristimulus values will appear to be a perfect visual match to the

standard human eye (which is supposed to represent the average of the entire

population with normal color vision). Since the x,y,z functions determine what

the calculated tristimulus values of any stimulus are, the functions are also

sometimes referred to as the color-matching functions .

With regard to opponent-color systems, the analogues of spectral tristimulus

values are also denoted by lower-case overbarred letters, as a, t, and d; or,

when the dependence of the functions on wavelength is to be emphasized , as aTx)

,

t(X), and d(X). These functions are thought of as the reponse sensitivities

113
of the three opponent-level mechanisms—luminance, red-green, and blue-yellow,

respectively.

Capital letters with overbars have no accepted meaning in colorimetry. In this

report, they have been used to symbolize vectors (see Note on Notation, p. xl).

114
NBS-IMA irev. 2-eo) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instruct/onsj

1. PUBLICATION OR

REPORT NO.

NBSIR/85-3202

2. Performing Organ. Report No, 3. Publication Date FEBRUARY 1986

4. TITLE AND SUBTITLE

Lln6ar Opponent-Colors Model Optimized for Brightness Prediction

5. AUTHOR(S)

Gerald L. Howett

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

national bureau of standards

DEPARTMENT OF COMMERCE

Gaithersburg, MD 20899

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

National Bureau of Standards - Department of Commerce

Gaithersburg, MD 20899 and

U.S. Department of Energy

Washington, D.C. 20585

10. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography gr literature survey, mention it here)

Formal nuliivariate optimization techniques were applied in an attenpt to determine how

well a linear, opponent-colors model of color vision could account for specific

brightness-matching data. The data fitted were from a single experiment by Sanders and

wyszecki that matched an adjustable white light in brightness to each of a set of

lights of 96 different colors and constant luminance. A generalized, linear,

opponent-colors model was formulated, which included the models of Guth (and

co-workers), Ingling (and co-workers), and Thornton as special cases. The model

ccxitained 10 parameters, including nine determining the spectral responses of the three

opponent-level channels and one determining the rule for coirbining the outputs of the

three channels to obtain an estimate of equivalent luminance (the luminance of an equally bright v^ite light) . Despite difficulties with the optimization procedure, a model was found that correlates better than 0.98 with the fitted data. The predictions

of this model for various other color-vision functions were explored and coirpared with

corresponding predictions of the Guth and Lodge model and the Thornton model. The new

model's predictions of these functions are less than perfect, but surprisingly good

considering that the model was c^imized entirely on brightness data (the only restriction being that the luminance channel should not have any negative values) . The model

was shown to predict the sort of conplex mixture of sub- and superadditivity that is

present in actual data. Some new algdDraic results concerning the "bA" or "B/L"

(eginvalent luminance over -luminance) ovals on the.x!bropaticity diagram were .dexived. —

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; ana separate key words by semicolons)

additivity; brightness; color; equivalent luminance; Guth model; luminance; model, color

vision; opponent colors; optimization, multivariate; primaries; white point

13. AVAILABILITY 14. NO. OF

Unlimited

1 1 For Official Distribution. Do Not Release to NTIS

PRINTED PAGLS

126

Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402. 15. Price

fy~l Order From National Technical Information Service (NTIS), Springfield, VA, 22161 $16.

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

USCOMM-OC I •
life

__

,i' ift’s^nS

Sal.--<;S

fj*

.-iXHtniiiq

oillowsa -uA $ma^'

3:

^ m^sm^^ ^

-^§nlk' a. acnt s^im b»:i^ii

' eiafiP 9<fT .aiab ' ialiL^ _

^'J9a' ; a ;-5o #j#& ' iti . 44gti| ©iiflir .:hmioam;jrniB'

^ita^sfioo/fcwa a:i:oXoo ' ai« to/

v^^aJ^ww<^^aai» "'XafeoiR^ «xo£os>-^"

xaioi^ (mmoifS Br» « CsciaAros^-oo brta) ^l^t; » (f

®d^' ^oVaattrtogpat laii.itwpi 'i©id^«xoq:4l,,.^

Ill© «(W, '^ni<4ao6^ aim M ^rlniMm:^^ ©no 6na . alomittfa^ Xsaatr—

na ad^> ^.flaWRrdw^ m iA$Mi od aX$wtfwifli>] a' '»ii* tt:ii» miM .(4d@>il

£^,6a^6X<|>ii»;' aspff'^ anoi^hsmil ^«>i^1^•*l£sle6' imU0 amiitisv '"Imx sJtdil

^ 9i$M Itm d^tS> adi' ls> im(>i:lQi:&#i<|.

£x>09 ss»x m aasiri:} to ^iwiiii^yiiiK'a^^

Kxti4s«W|;f|gp' Mob (ffmff&^M: m IbowJb^^ iatwar

srtIJb^''

blooda Xamutdo do^mliKuX ipl JliTHI#

'©iti ^i^ttkmmm

oXmiie^lavwafi' *ii|jing ilifiwh

'»««XSv|ypp^:';'aoioa

ll ,vgli* if''

;f4

:^44.

4ic

., n* , wr-,,

'v|'^, ;»w«»^»i*«<a^i*i9|»0;^^^

^,W|ill|«l.»«i , ,iailTr'*0 *Wyi»e ; rtoiiBinweffol liw4«»*ii^T iitmlimilll iWr*^''.t*»«0 {jQ;

-‘VsSS^